RELATION ALGEBRAS
STUDIES IN LOGIC UND AND FOUNDATIONS OF MATHEMATICS MATHEMATICS THE FOUNDATIONS VOLUME 150
Honorary Editor: P. SUPPES
Editors: M G . ABRAMSKY, A B B A Y , London London S. S. ARTEMOV, Moscow D.M. DM. GABBAY, London A. A. KECHRIS, Pasadena A. PILLAY, Urbana A. R.A. SHORE, Ithaca
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AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE •. SYDNEY •. TOKYO
RELATION ALGEBRAS
Roger D. MADDUX Department of Mathematics Iowa State University Ames, Iowa 50011 USA USA
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AMSTERDAM •. BOSTON •. HEIDELBERG •. LONDON •. NEW YORK •. OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
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Preface This book is an introduction to the calculus of relations and the theory of relation algebras. It covers a period stretching from the middle of the nineteenth century to the end of the second millennium. George Boole [33, 32] turned the logic of classes, propositions, and probabilities into algebra, thereby creating a calculus of classes which, by the end of the nineteenth century, was codified as the class of algebras now known as Boolean algebras (a name suggested by H. M. Sheffer in 1913; see Huntington [105, p. 278]). Starting shortly after Boole's pioneering work, Augustus De Morgan [63, 64, 65] introduced binary relations to logic. He defined various operations on binary relations and stated many of their algebraic laws, such as his "Theorem K," which he derived from his analysis of generalized Aristotelian syllogisms. By applying the algebraic methods of Boole to the relations of De Morgan, Charles S. Peirce [191, 192, 193, 189, 195, 194, 196, 198] created a calculus of binary relations, as well as calculi of relations of higher rank. Peirce considered and analyzed the properties of a wide variety of operations on relations. Peirce's calculus of binary relations, formulated in his short paper Note B [194] of 1883, was very extensively developed by Ernst Schroder [214, 215] in the third volume of his Vorlesungen iiber die Algebra der Logik, published in 1895. Schroder invented the notational system used in this book for the operations in a relation algebra. In 1915 Leopold Lowenheim [131] proved that if an equation of the PeirceSchroder calculus of binary relations fails in some domain but holds in all finite domains, then it must fail in a countably infinite domain. Lowenheim's theorem, with improvements and extensions by Skolem [218, 219, 220], Tarski [221, p. 161], [238, p. 568], and Mal'cev [159], became the Lowenheim-Skolem-Tarski theorem (see Th. 172 below). This theorem is now a cornerstone of model theory, a subject that came into existence many years after Lowenheim's paper was published. It is in Lowenheim's 1915 paper that the expression "first-order", as in "first-order logic", makes its first appearance. For a while the Peirce-Schroder calculus of relations was overshadowed by the rise of the logistic movement, stemming from the work of Prege, Peano, Whitehead, and Russell. For a more complete account of this history, see Lewis [129], Tarski [226, pp. 73-74], Moore [185], Vaught [248, 249, 250], Merrill [170, 171, 172, 173], Tarski-Givant [240, p.xv-xix], Anellis-Houser [13], Brady [35], Badesa [16], and van Heijenoort [247] (which includes English translations of the papers by Lowenheim and Skolem).
viii
PREFACE
In his first paper on the subject, published in 1941, Tarski [226] outlined different ways to set up rigorous deductive foundations for the Peirce-Schroder calculus of binary relations. His first method is to extend first-order predicate calculus of binary relations with symbols for operations on binary relations and a symbol for equality of binary relations, together with additional axioms (besides the usual ones for first-order logic) that express the meanings of these new symbols. This extended system contains equations such as R + S = S + R, which expresses the commutativity of union. In his second method, Tarski used only Boolean combinations of equations and proposed an entirely new set of axioms. A briefly mentioned third variant contains only equations. The formulas provable in the second and third methods may now be simply described as the ones true in all relation algebras, while the formulas provable by the first method are the ones true in all representable relation algebras. In the previous year J. C. C. McKinsey [168], addressing Tarski's problem of formulating the theory of relations as an abstract mathematical system as had been done for the calculus of classes, found axioms that characterize what may now be described as the class of simple complete atomic representable relation algebras. After proving several theorems according to his second method, Tarski [226, p. 88-89] discussed several metalogical problems connected with his proposed foundations and announced two new theorems. One was an extensive generalization of the result due to A. Korselt (presented in Lowenheim [131]) that not all first-order definable properties of relations can be expressed as equations in the calculus of relations. The other may be described as the undecidability of the equational theory of representable relation algebras. Tarski planned to publish proofs in a separate paper, and started writing a manuscript in the summer of 1942 (Feferman-Feferman [71, p. 152]), but by then he had discovered some much deeper and far more significant results. In his 1915 paper Lowenheim had written, "All important problems of mathematics and of the calculus of logic can, it seems, be reduced to such relative equations" [247, p.233], and he expanded on that theme in a paper published in 1940 [132]. Lowenheim may well have been working on that paper when he first met Tarski in 1938 at the home of Kurt Grelling in Berlin, where they discussed logic for hours (Feferman-Feferman [71, p.103]). Tarski subsequently carried out Lowenheim's suggestion to "Schroderize" mathematics ("Auch die Cantorsche mengenlehre lasst sich verschrodern", Lowenheim [132, p. 2]) by proving that practically every axiomatized set theory can be formulated as a set of equations in the calculus of relations, and that every set-theoretical proof can be converted into an equational derivation from these equations. The equational axioms for set theory contain no variables ranging over sets and only a single constant that represents the membership relation. A consequence of this result is the undecidability of the calculus of relations formulated according to Tarski's second and third methods, i.e., the equational theory of relation algebras is undecidable. When Tarski visited the Institute for Advanced Study in early 1942, conversations with Kurt Godel led to several applications, including a simplified axiom set for von Neumann-Bernays-Godel set theory. By March 1942 Tarski had also found an undecidable fragment of sentential calculus (see Tarski-Givant [240,
PREFACE
ix
p. 168]). The long manuscript on the calculus of relation written in 1942-1943 (Tarski [227]) was never published in its original form. The main results were announced in 1953 in three abstracts, Tarski [230, 229, 231]. Tarski started revising the manuscript in 1971. It eventually appeared four years after his death as Tarski-Givant [240]. The main result of the Tarski-Givant book and Tarski's manuscript is the "Main Mapping Theorem for £x and £ + " (see Tarski [227, 5.22], Tarski-Givant [240, 4.4(xxxiv)], and (7.187) below), which says that if a theory, formalized in the first-order predicate calculus, proves the existence of a pair of functions acting sufficiently like projection functions (which map ordered pairs to their components), then that theory can also be formalized in the calculus of relations. This gives, as a particular application, Tarski's formalization of set theory without variables (see Tarski [227, 7.3], [229], and Tarski-Givant [240, §4.6]). As a further application of the Main Mapping Theorem for Cx and C+, Tarski was also able to prove his QRA theorem (see Tarski [230, VII], TarskiGivant [240, 8.4(iii)], and Th.427 below). The QRA theorem asserts that if a relation algebra contains a pair of quasi-projections (elements that behave like projection functions) then it is representable. It was presumably proved between 1943 and 1952, since it is does not appear in the 1943 manuscript. The theory of relation algebras was extensively developed by Tarski and his colleagues and students in the 1940's and 1950's. The algebraic systems discussed by Tarski [226] in 1941 coincide with what we may now describe as simple relation algebras. A definition of relation algebras equivalent to the current one was first published in 1948 in the abstract J6nsson-Tarski [117]. In his 1941 paper, Tarski [226, p. 87-88] asked whether his axioms (given on p. 19 below) are complete, that is, whether every sentence in the calculus of relations that is true in every algebra of binary relations can be derived from his axioms. A closely related question, also asked by Tarski in that paper, is whether every model of his axioms is isomorphic to an algebra of binary relations. In 1950, R. Lyndon [133] answered both questions by constructing models of Tarski's axioms that are not representable. It follows that Tarski's axioms are incomplete. Many significant general algebraic theorems concerning relation algebras were proved by J6nsson-Tarski [118, 119], For example, they proved that every relation algebra 21 can be embedded in a complete atomic relation algebra, called the perfect extension of SI. Every positive equation (one that does not involve complementation) that holds in a relation algebra also holds in its perfect extension. Relation algebras can be characterized by positive equations, so the perfect extension of a relation algebra is again a relation algebra. In 1955 Tarski [232] proved that the class of representable relation algebras is an equational class, that is, it can be defined by a set of equations. In 1956 Lyndon [134] produced an explicit equational basis for the class of representable relation algebras. Jonsson-Tarski [119] proved that every relation algebra has a representation in which complementation and intersection need not behave properly. In 1959 Jonsson [112] also characterized those relation algebras that have representations in which complementation and union need not behave properly. In the same paper
x
PREFACE
Jonsson constructed a nonrepresentable relation algebra from a non-Desarguesian projective plane. In 1961 Lyndon [135] altered Jonsson's construction to establish a strong link between projective geometries and a certain class of relation algebras. In 1964 J. D. Monk [176] used this connection to prove that the class of representable relation algebras has no finite equational axiomatization. Tarski and McKinsey observed (see Jonsson-Tarski [119]) that every group 85 gives rise to a relation algebra in a very natural way—consider the Boolean algebra of subsets of -verse of as,' read as '$-verse x.' Relations are assumed to exists between any two terms whatsoever. If X be not any L of Y, X is to Y in some not-L relation: let this contrary relation be signified by I; thus X.L Y gives and is given by X..IY. Contrary relations may be compounded, though contrary terms cannot: Xx, both X and not-X, is impossible; but LIX, the L of a not-L of X, is conceivable. Thus a man may be the partisan of a non-partisan of X. (De Morgan [66, pp. 222-223]) Peirce introduces complementation and conversion as follows. The formulas he lists have been converted into our current notation and numbered for later reference. The letters R and 5 replace Peirce's letter I and 6, the inclusion symbol C is used instead of Peirce's symbol -*
... It is to be remarked that S,a;, and IT,a;, are only similar to a sum and a product; they are not strictly of that nature, because the individuals of the universe may be innumerable (Peirce [199, 3.393].) Peirce and Mitchell share credit with Frege for the creation of quantifiers (see Church [50, fn. 103, p. 45, fn.453, p. 288], Moore [185], and Brady [35]). Peirce [199, 3.499-3.509] gives another brief description of his general algebra of logic, with specific reference to quantifiers. Later works in this tradition are WhiteheadRussell [255], Lewis [129], and Lewis-Langford [130, pp. 104ff]. Tarski's first approach (founding the calculus of relations within first-order predicate calculus) is worked out in detail by Tarski-Givant [240] and Chapter 7. However, in 1941 Tarski thought that
8. AXIOMATIZATION OF THE CALCULUS OF RELATIONS
19
this method has certain defects from the point of view of simplicity and elegance. We obtain the calculus of relations in a very roundabout way, and in proving theorems of this calculus we are forced to make use of concepts and statements which are outside the calculus. It is for this reason that I am going outline another method of developing this calculus. (Tarski [226, p. 77]) Tarski therefore proposed a second method, one which only axioms that are specific to the calculus of relations. Starting with an axiomatization of the sentential calculus, Tarski adds axioms (I)-(XV) below. The symbols +, , ;, f) a n d are operation symbols whose intended interpretations are the operations U, fl, , f, and ~1, respectively. The symbols 0, 1, 1', and 0' are constants, V means "or", A means "and", => means "implies", ~ means "not", and, of course, the symbol = is the equality symbol, denoting the identity of the relations denoted by expressions on either side of it. The labels for these axioms are the same ones used by Tarski. (I)
(II) (III) (IV) (V)
(VI) (VII) (VIII) (IX) (X)
(XI) (XII) (XIII) (XIV)
(R = S A R = T) => S = T, R = S => (R + T = S + T A R R +S =S +R A R S= S R , (R + S ) - T = R - T + S - T A (R S) + T = (R + T) (S + T),
R + 0 = R A Rl = R, R + R=l A RR = O, ~ ( l = 0), R = R, (R;Sy = S;R, R;(S;T) = (R;S);T, R;V =R, R;l = l V l;fl= 1, (R;S)-f 0' = F ,
= O => (S;T)-R
= O,
(XV) By the way, both Schroder and Tarski invented a special symbol to be used in place of the symbol f, a practice not followed here. Instead, we use the symbol f originally chosen by Peirce. The only rules of inference are the rule of substituting equals for equals and the rule of detachment (modus ponens). Axioms (I)-(VII) are an axiomatization of the calculus of classes, due essentially to E. V. Huntington [103, 102] (see Tarski [226, p. 78, fn. 3]). The remaining axioms are selected from
20
1, CALCULUS OP RELATIONS
laws already noted by De Morgan, Peirce, and Schroder. In particular, (VIII) is (1.5), (IX) is part of (1.37), (X) is (1.23), (XI) is part of (1.50), (XII) expresses part of (1.54), (XIII) is an alternative expression of part of (1.14), (XIV) defines 0', and (XV) defines f- Tarski [228] proves sixteen theorems in this system, many of which are among the formulas already listed. Tarski then proves the theorem below, which was originally proved for binary relations by Schroder [215, §11, pp. 153fi]. By a sentence, Tarski means a Boolean (or sentential) combination of equations. The formulas in the proof below have the form A O- B, an abbreviation of (A => B) A {B A). Tarski's proof is based just on the axioms (I)-(XV), so what he proves is a strict generalization of Schroder's theorem; see Th. 381 below. Theorem 1 (Sehroder-Tarski Theorem). On the basis of axioms (I)-(XV), every sentence is provably equivalent to an equation of the form "R = 1". PROOF. The theorem follows (by induction) from the following formulas, all of which are easily derivable from Tarski's axioms (I)-(XV). (1.55)
R =S
(1.56)
R-S + ~R-~S = 1
~ (R = l) «* 1;S;1 = 1
(1.57)
R=lhS
(1.58)
R = 1VS = 1 « Oti*tOt#tO = l
(1.59)
=l
(R = 1 =s- S = 1)
R-S = l 1;1;1 + 5 = 1
Either one of the following two formulas can be used in place of (1.58): (1.60)
fl=lVS
= l «
(1.61)
fl=lVS
= l «
Tarski had a high opinion of Schroder's book, which, he said, "contains a wealth of unsolved problems, and seems to indicate the direction for further investigations" (Tarski [226, p. 74]). Peirce was not so enthusiastic, as shown by his brief historical summary. Although Schroder developed Peirce's algebra from Note B, Peirce thought that "Professor Schroder attaches, it seems to me, too high a value on this algebra." (Peirce [199, 3.498]). For further explication of Schroder's work by Peirce, see Peirce [198], [199, 3.510-3.525], and Brady [35]. 9. Definitions of relation algebras The similarity type of the algebras in RA (the class of relation algebras) and the defining characteristics of those algebras have evolved. Tarski [226, p. 87] mentions the possibility, based on the Schroder-Tarski Th. 1, of an equational foundation for the sentential calculus of relations. Implicit in this plan, which "has not been worked out in detail" but "presents no essential difficulty", is (what became) an equational definition of relation algebras. The situation is similar to that of groups. Indeed, Tarski noted "that the calculus of relations includes the elementary theory of groups and is, so to speak, a union of Boolean algebra and
9. DEFINITIONS OP RELATION ALGEBRAS
21
group theory." The class of groups forms a variety, because it is closed under the formation of subgroups, homomorphic images, and direct products. These closure properties can be made obvious from a general algebraic point of view by defining groups with only equations. For this purpose it is convenient to consider groups as algebras that have a binary operation o, a unary operation - 1 , and a distinguished element e. Then a group is an algebra that satisfies these equations: (x o y) o z = X o (y o z), eox = x = xoe, x~x
0 1 = 1 0 x~l
=
e.
By the mid-1950's Tarski defined relation algebras similarly, using purely equational postulates. See, for example, Tarski [232, p. 60] and Tarski [233, §3]. An equational definition is also adopted in this book, following Tarski-Givant [240, Def. 8.1]. A relation algebra is an algebraic structure of the form 21 = (A,+,~, ;,",1'>, consisting of a set A, two binary operations on A, namely + and ;, two unary operations on A, namely ~~ and ", and a distinguished element 1' 6 A, such that - {A, +,~) is a Boolean algebra (by axioms R1-R3 below), - the binary operation ; is associative (axiom R4) and distributes over union from the right (axiom R5), - the element 1' is a right identity element for ; (axiom Re), - the unary operation " is idempotent (axiom R7), distributes over union (axiom Rg) and satisfies a law from group theory (axiom Rg), - the algebra satisfies axiom Rio. Here are those ten equational axioms: Ri
R2
R3 R4 Rs
x + y = y + x, x-+- (y + z) = (x + y) + z, x + i7+ x + y = x, x;(y;z) = (x;y);z, ([x + y ) ; z = x ; z + y ; z ,
Re
x;V
R7
X =
X,
(x + y)" = x + y, (x;y) = y\x,
Rs Rg
Rio
= x,
x
-\-y =
y.
+-commutativity +-associativity Huntington's axiom ;-associativity ;-distributivity identity law "-involution "-distributivity "-involutive distributivity Tarski/De Morgan axiom
The first three axioms R1-R3 are enough to ensure that (^4, +, ) is a Boolean algebra. They are due to E. V. Huntington [105, 104, 106]). Axioms R 4 R10 were chosen by Tarski. Many of these equational axioms have appeared in Peirce's work; see (1.23) for R4, (1.26) and (1.28) for R5, (1.50) for R6, (1.5) for R7, (1.36) for Rg, and (1.37) for R9. Other operations and elements, such as
1. CALCULUS OF RELATIONS
the ones corresponding to intersection, difference, empty relation, and universal relation, can then be defined by (1.62)
x-y:=x
+ y,
(1.63)
x-y:=x
+ y,
(1.64)
0:=l'+r,
(1.65)
1:=1'+1\
In 1948 Jonsson-Tarski [117] defined a relation algebra as a Boolean algebra together with a binary operation ; and a unary operation ", such that R4-R5, R7-R10 hold, and R6 holds for some element 1'. The same definition was adopted by Chin-Tarski [49, Def. 1.1]. All the postulates are equations except the one asserting the existence of an element 1' satisfying R6. In the presence of the axioms listed before it, Rio is equivalent to De Morgan's Theorem K. Equation Rio does not occur in Peirce's Note B, but it can be replaced by either (1.24) or (1.25), if the occurrences of f in those formulas are eliminated according to (XV). The language that Tarski [226] uses for his axiomatization of the calculus of relations includes variables R, S, etc., denoting arbitrary binary relations, symbols for the distinguished binary relations 0, 1, 1', and 0', symbols for the six operations of Note B, namely +, , ;, f, ", and ~, a symbol for equality, and propositional connectives. If 21 is a model of Tarski's axioms (I)-(XV), where 21 is an algebraic structure that is appropriate for this language, say
%={A, +,-,", 0,1, ;,f, V , 0 ' } , then it follows from (I)-(VII) that the reduct (A, +, , ~, 0, l) is a Boolean algebra with at least two elements. Using (I)-(XI) and (XIII) it can also be shown that the reduct (A, +, , ~, 0,1, ;, ", 1') is a relation algebra as defined here (see Chin-Tarski [49, p. 352] and Jonsson-Tarski [119, p. 128, fn. 15]). Conversely, if (A,+,~, ;,",1') is a relation algebra and 0', f, , 0, and 1 are defined by (XIV), (XV), (1.62), (1.64), and (1.65), respectively, then the algebra 21= (A, + , - , " , 0,1, ;, f, ", 1', 0'} is a model of (I)-(VI), (VIII)-(XI), and (XIII)-(XV). Furthermore, 21 is a model of (VII) just in case 21 has at least two elements. Finally, 21 is also a model of (XII) just in case 21 has exactly two nontrivial congruence relations, namely, the identity relation on A and the universal relation on A. An algebra is said to be trivial iff it has exactly one element. A trivial algebra has only one congruence relation, but an algebra with at least two elements always has at least two congruence relations, namely, the identity relation and the universal relation. A nontrivial algebra with exactly two congruence relations is said to be simple. Every simple relation algebra yields a model of Tarski's axioms (I)-(XV). If relation algebras were defined as models of (I)-(XV), the resulting class of algebras would not be a variety, for it would not be closed under the formation of homomorphic images because of axiom (VII), nor closed under the formation of direct products because of axiom (XII). If axioms (VII) and (XII) are deleted, then the remaining ones form an alternative axiomatization of relation algebras, in which the identity element 1' is treated as a
9. DEFINITIONS OF RELATION ALGEBRAS
23
distinguished element and all axioms are equations except axiom (XIII), which is an implication. Axiom (XIII) is an alternative form of De Morgan's Theorem K. Lyndon [133, p. 708] defined a relation algebra as a Boolean algebra together with a unary operation, a binary operation, and a distinguished element, satisfying (VIII)-(XI) and (XIII), with (XII) replaced by a variant of (1.56), namely
He does not include (VII), so Lyndon's relation algebras are actually simple or trivial (having only one element) relation algebras. Lyndon's definition is not only not equational, but is not equivalent to any equational definition. Some of the results of Jonsson-Tarski [119] require that complementation cannot be obtained from the fundamental operations by composition, so they do not include complementation in the similarity type of relation algebras. See Jonsson-Tarski [118, p. 897] for further comments on this situation. Jonsson and Tarski [118, Def. 4.1] define a relation algebra as an algebra of the form 21 = {A, +, 0, , 1, ;, ", 1'), where (A, +, 0, , 1) is a Boolean algebra, satisfying R4, R,6, and a version of De Morgan's Theorem K, namely, x;y
z = 0 x ; z
y = 0 z ; y
x =0 .
This definition of relation algebras is equivalent to the others given above (unless an axiom guaranteeing simplicity is included). Several other variations on these definitions are possible; see, for example, Chin-Tarski [49, p. 352, fn. 10, p. 354, fn. 12], Jonsson-Tarski [119, p. 128, fn. 15], and Brink [36, 39]. For related results see Andreka-Comer-Nemeti [7], Borner [34], Diamond-McKinsey [67], Andreka [6]. For example, this last paper shows that no equational axiomatization of RA is possibly using only equations that contain a single variable. The axiomatic relation algebraic approach to the calculus of relations has two important features that distinguish it from the viewpoint of Peirce and Schroder. First, Peirce and Schroder work in 9\e (U), the square relation algebra consisting of all binary relations on the universe of discourse U (see definition (3.76)). But Tarski's axioms hold in models formed by taking only some of those relations. Indeed, a model of (I)-(XV) can be obtained from any set of relations on a set U that contains the identity relation on U and is closed under union, complementation with respect to U2, relative multiplication, and conversion. In other words, any subalgebra of $He (U) is a model of Tarski's axioms. In such models, Peirce's "curious development formulae" (1.30)-(1.33) make sense, but may not hold. For example, if U has two or more elements, then 0, 1, 1', and 0' are distinct relations that form a 4-element subalgebra of D\c(U) and model of (I)-(XV). If U has exactly two elements then 0'; 0' = 1', while if U has three or more elements then 0';0' = 0'. Suppose U has three or more elements. Then Peirce's formula (1.30) fails when x = 0',y = V, and z = 1, because (x-y);z = (0' l');l = 0;l = 0 but
1, CALCULUS OP RELATIONS
= (o +1) (i +o) (0'+ 0') (o! + r) = 0' # 0. However, Peirce made no mistake, for (1.30)-(1.33) do hold in 9te(J7). For a proof, see Schroder [215, §29, pp. 491^94]. The second distinguishing feature of the relation-algebraic approach to the calculus of relations is that there are relation algebras of binary relations in which the universal relation need not relate every element of the universe of discourse to every other. One way to define such algebras is to slightly modify the definition of complementation in (1.2) to make it suitable for the case in which 1 is not a relation of the form ?72. Specifically, let (1.66)
x := {{a, b) : {a, b) £ 1 and (a, &} £ x}.
Then a proper relation algebra is an algebra SI = {A, +,~, ;, ", 1'} whose universe A is a set of binary relations, all contained in some largest binary relation 1 € A, such that + is union, ~ is complementation with respect to 1, ; is relative multiplication, w is conversion, 1' is the identity relation on the field of the largest relation 1, A is closed under +, ~, ;, and ", and 1' € A. It is straightforward to check that every proper relation algebra is a relation algebra. It is also easy to show that the relation 1 in a proper relation algebra must be an equivalence relation. (By closure under relative multiplication, l j l g A , hence 1;1 C 1, so 1 is transitive. Similarly, 1 E A, hence 1 C 1 , hence 1 is symmetric.) If A is the set of all subrelations of an equivalence relation E, then A obviously has the properties required for St to be a proper relation algebra. In this case St is called an equivalence relation algebra. It is a model of Tarski's axioms (I)-(VI), (VIII)-(XI), and (XIII)-(XV). » will also be a model of (VII) if E is not empty, and it will be a model of (XII) if E is has a single equivalence class. All of its subalgebras are also proper relation algebras. A relation algebra is said to be representable if it is isomorphic to a proper relation algebra. RRA is the class of representable relation algebras. In particular, square relation algebras and equivalence relation algebra are representable. J6nsson-Tarski [119] do not require a proper relation algebra to contain the identity relation on its underlying set. According to their definition, J6nssonTarski [119, Def. 4.23], a proper relation algebra is required to be a relation algebra, so there must be a relation that acts as an identity element for relative multiplication, but they do not require this relation to be an identity relation. It is easy to show, from some of the axioms for relation algebras, that the identity element does have to be an equivalence relation. By J6nsson-Tarski [119, 4.27] (see Th. 102), a relation algebra is isomorphic to a proper relation algebra in the sense of Jonsson-Tarski [119, Def. 4.23] iff it is isomorphic to a proper relation algebra in the sense adopted here. The definition adopted here also occurs in Jonsson-Tarski [117] and Tarski-Givant [240, §8.3].
11. INCOMPLETENESS
25
10. Undecidability and inexpressibility Already in 1883 Peirce made a prophetic observation regarding the algebraic system that he had invented. The logic of relatives is highly multiform; it is characterized by innumerable immediate conclusions from the same sets of premises. ... The effect of these peculiarities is that this algebra cannot be subjected to hard and fast rules like those of the Boolian calculus; and all that can be done in this place is to give a general idea of the way of working with it. (Peirce [195, pp. 192-193], [199, 3.342]) Tarski confirmed Peirce's evaluation by proving that there is no algorithm for determining whether an equation in the calculus of relations can be derived from a given set of premises. This result, first announced in Tarski [226, p. 88], yields the undecidability of the equational theory of representable relation algebras; see Tarski-Givant [240, §8.7, p. 268]. Tarski posed two other questions. Is every sentence in the first-order predicate calculus of binary relations equivalent to (expressible as) an equation in the calculus of relations? Alwin Korselt had shown years earlier that the answer to this question is "no", and his result was reported by Lowenheim [131, Th. 1] (van Heijenoort[247, p. 233]). Tarski repeated Korselt's example, gave another one, and announced a far-reaching generalization of Korselt's result. Tarski's generalization was included in the 1943 manuscript [227], published as Tarski-Givant [240, 3.5(viii)], and proved again by Jonsson [115, 1.5.3]. Tarski also asked whether there is an algorithm for determining whether a first-order sentence is expressible as an equation, but thought it "seems plausible that the answer is negative" (Tarski [227, p. 158], [226, p. 89]). In the early 1970's M. Kwatinetz [126] showed that there is indeed no such algorithm. 11. Incompleteness Tarski's axioms (I)-(XV) suffice to prove very many formulas stated by De Morgan, Peirce, and Schroder, so it is natural to ask whether this axiomatization is complete. It is not, as was shown in 1950 by results of R. Lyndon discussed in the next section. In 1941 Tarski [226] put the problem this way. Is it the case that every sentence of the calculus of relations which is true in every domain of individuals is derivable from the axioms adopted under the second method? This problem presents some difficulties and remains open. I can only say that I am practically sure that I can prove with the help of the second method all of the hundreds of theorems to be found in Schroder's Algebra und Logik der Relative. (Tarski [226, pp. 87-88])
26
1. CALCULUS OF RELATIONS
Clearly Tarski [226] did not explicitly conjecture that his axioms (I)-(XV) are complete, but in the 1943 manuscript he was less reserved. In the quotations that follow, "C-formula" means a formula in the calculus of relations; axioms (I)-(XV) and the formulas (1.55)—(1.61) are good examples of C_-formulas. An "R-formula" is simply an equation; axioms (VIII)-(XI), (XIV), and (XV) are examples of Rformulas. C-formulas may be described as Boolean (or sentential) combinations of R-formulas. An "L-formula" is a formula in a first-order predicate calculus of binary relations that includes all the relation terms that are used to build C_formulas. For example, Vx3y(x(R- S) +Ty) is an L-formula. Thus all R-formulas are C_-formulas and all C-formulas are L-formulas. A C-formula is "C_-provable" if it can be proved from the axioms (I)-(XV) using the rule modus ponens. An L-formula is "L-provable" if it can be proved using the full power of first-order logic. Tarski formulated the completeness problem as follows. Problem 2. Is it true that every C-formula * which is Improvable is also C-provable? (Tarski [227, p. 1-5]) By the Completeness Theorem of Godel [83], "L-provable" is equivalent to "true in every 9te(E/)". Hence, Problem 2 may be reformulated in the following way. Problem 2'. Is it true that every C_-formula \I/ which is generally satisfied in every universe of discourse is also C-provable? This problem still remains open. Therefore the whole problem of axiomatization of the relation calculus cannot be regarded as definitely settled: For, should it turn out that the solution of the problem is negative, we should have to look for an extension of the present system of C_-axioms so as to secure the desired completeness property. It seems, however, very plausible that the solution of Problem 2' is positive—if only for this reason that all the numerous Improvable C-formulas which are known at present have turned out to be C_-provable; and the results in Chapter 5 seem to support this supposition. In particular, we shall see that the solution is indeed positive, if instead of L and C_ we consider the enriched theories Li andC_i: every Li-provable C_i-formula is Ci-provable. (Tarski [227, p.I-10]) The subscript "1" indicates the introduction of two new relation constants. The relations denoted by these constants can be called together conjugated ordered couple relations, for they are assumed to establish a one-to-one (or even one-to-many) correspondence between all ordered couples formed from the elements of the universe U on the one hand,
11. INCOMPLETENESS
27
and the elements of U, or a subset of U, on the other. (Tarski [227, 1-9]) At various points later in the manuscript, Tarski makes statements similar to the remarks published in Tarski [228]. A rough outline of an algebraic development of the relation calculus will be given in Chapter 8. Here we want only to remark that all theorems of this calculus that are known at present can be derived from a small number of formal laws, which— in an algebraic presentation of the calculus—can be taken as postulates to define the notion of the relation algebra. (Tarski [227, p. 17]) For the problem whether every ^-formula which is intuitively true is (^-provable still remains open. Furthermore, the methods of proof used in the calculus Q are by no means familiar; and therefore even the derivation of the most elementary formulas may cause some difficulties. We believe, however, that a reader who becomes interested in the subject will rather easily acquire the necessary skill which will enable him to deduce from C_-axioms all (^-provable formulas that are involved in later developments. (Tarski [227, p. 132]) Theorem 5.22, referred to in the next quotation, is the Main Mapping Theorem for £ x and £+ (see p.ix, (7.187), or Tarski-Givant [240, 4.4(xxxiv)]). As was mentioned in the introduction, the question whether 5.22(i) holds for every C-formula # (and not only for formulas of the type KA,B —> $), i.e., whether every L-provable (3-formula is C_-provable, still remains open; and the same applies to the apparently more general, but actually equivalent, questions whether 5.22(ii) holds for every class of G-formulas (and not only for those classes which contain KA,B)- It seems very probable that the answer to these questions is affirmative; at any rate, as was pointed out in the introduction, all the numerous L-provable C_-formulas which are found in the literature can be shown to be G-provable. While the result obtained in 5.22 makes an affirmative answer still more plausible from an intuitive point of view, it does not seem to bring us closer to the actual solution of the problem. If we succeeded in attaining this solution, Theorem 5.22 would clearly lose all importance. This was the main reason why we refrained from giving here the tiresome proof of Lemma 5.18 upon which that of 5.22 is based.
28
1. CALCULUS OP RELATIONS
Theorem 5.22 did not "lose all importance", because R. Lyndon [133] proved the existence of an L-provable C-formula that is not (^-provable. This made the "tiresome proof of Lemma 5.18" unavoidable in the proof of Tarski-Givant [240, 4.4(xxxiv)]). Tarski continued, On the other hand, we want to state here without a detailed proof a result which seems to make a real contribution toward the solution of the problem in which we are interested. (Tarski [227, p. 169-170]) This result (Tarski [227, Th. 5.24, p. 170]) is the equipollence of the relation calculus C_ with a form of 3-variable logic. A statement of Theorem 5.24 and a description of its proof occupy only two pages of the manuscript. The details involved in working out both a correct formulation of this theorem and complete and correct proof were one of the sources of delay in the completion of TarskiGivant [240]; see the historical remarks in Tarski-Givant [240, pp. 88-89]. Theorem 5.11, mentioned in the next quotation, is a precursor to TarskiGivant [240, 4.4(xiv)]. An approximate statement of this result is that, relative to a particular sentence asserting that two binary predicates are quasi-projections, every L-formula can be converted (by a well-defined recursive procedure) into an R-formula (an equation) with which it is L-provably equivalent. We are now going to derive from our fundamental theorems 5.11 and 5.22 an important consequence. We shall show that— in spite of the fact that not every L-formula can be transformed into an equivalent CJ-formula and that the problem whether every L-provable C-formula is C_-provable still remains open— every problem regarding the L-provability of an L-formula reduces to that of the C-provability of a suitably constructed CJ-formula. Speaking more precisely, we shall correlate with every L-formula $ a C-formula $* (which is constructed from $ in a finite number of well-determined steps), and we shall show in 5.31 that the CJ-provability of e = i V e = 2)
AVe(e £ r
> e = j ) V e = ? ) A r £ C)).
Recall that Tarski's S3 expresses the existence of A-1\B. This long form differs from Tarski's S3 only in the placement of a few variables. Are (2.101) or S3 equivalent to significantly shorter sentences, or can they be replaced, in the context of other axioms, with substantially simpler axioms? We may now define the unit relation V2 as simply the relative product of the universal class with itself. V2 := V|V.
(2.102) Then V2 =
{(x,V):x,yeV}.
A class R is a binary relation iff it is a class of ordered pairs iff R C V2, and R is a binary relation on the class A iff R is a binary relation and the left and right sides of every ordered pair in R are elements of A, that is, Vc(c £ R => 3^(1 £ A A 3j,(p 6 A A c = (m,y)))). For every class R, R(~\V2 is called the relational part of R. It is the largest binary relation included in R. Note that A\B and A f B are always binary relations, although they are not necessarily sets. Theorem 15. Let R and S be arbitrary classes. (i) The relative product R\S and the relative sum R^ S exist and are binary relations: (2.103)
R\S C V2,
(2.104)
RjSCV2.
(ii) R is a binary relation » R C V2 « R C R\V « R C V|i2, (iii) V2j iZnV 2 , and 9 are binary relations. Various elementary results concerning relative sums and products are gathered in the following theorem. The notation is conventional—unary operations take precedence over binary ones. Among the binary operations, the order of precedence is |, f, n, U, and then ~ . For repeated binary operations, use association to the left, so that, for example, AUBUC = (AUB)UA, Several parts of the
11. AXIOM OF RELATIVE PRODUCT
Bl
following theorem have been given descriptive names or labels for later reference. For example, "|-mon" stands for "monotonicity of |". Theorem 16. Let R, S, T, X, Y, Z be arbitrary classes. The universal class may be used in place of the unit relation. (2.105)
R\S =(Rf] V2)\(S n V2) = (RC\ V2)\S = R\(S n V2),
(2.106)
R\\/ = R\\/2,
(2.107)
\/\R = \/2\R,
(2.108)
R^S = (i?nv2)t(5*nv2) = (RnV2)fs = Ri(snv2),
(2.109) (2.110) Relative multiplication is normal, and relative addition is dual-normal. (2.111)
0|fl = 0 = .R|0,
(2.112)
M]R = V2 =R\M.
Laws of properties.
(2.113)
s | r n v | i ? = s|(v|i?nr),
(2.114)
R\\/nS\T=(SnR\\/)\T,
(2.115)
i?|VnV|S = i?|V|5',
Schroder's Law [215, p. 150,152].
Tarski's Law [226] R\V = V2 Peirce's Remarkable Property [194]. 01-Rt 0 "
either
0 or V2,
(0f-R)|V is either 0 orV 2 , 0f-R|V is eitfjer 0 orV 2 , V|i?|V is eitter0 or V2. (|-assoc, f-assoc) Relative multiplication and relative addition are associative. (2.116)
fllOSiT)
= (R\S)\T,
(|-dist) Relative multiplication distributes over union. R\(SUT)
= R\SUR\T,
(SUT)\R
=
S\RUT\R.
(f-dist) Relative addition distributes over intersection. (2.118)
2. SET THEORY
(2.119)
(S
(duality) Relative multiplication and relative addition are dual. 2
(2.120) (2.121)
(|-mon, f-tnon) Relative multiplication and relative addition are monotonic. (2.122)
R C 5 =4- R\T C S\T,
(2.123)
R C S =j- T|i? C T|S,
(2.124)
ECS =s- EfTCSfT,
(2.125)
E C S =s-
TjRCTjS.
(Peirce's Law) "Two formulm so constantly used that hardly anything can be done without them" Peirce [194]. (2.126)
R\(S^T)CR\S^T,
(2.127)
(RjS)\T
C Rf S\T.
Miscellaneous interesting laws. (2.128)
R\Sn(R}T)=R\(SnT)n(RjT),
(2.129)
R\Sn(R1[T)=R\(S~T)n(R^T),
(2.130) (2.131)
R\Sn(X^Y) C(RnX)\SUR\(SnY), R\s\Tn(x^YfZ) c
One of the "miscellaneous interesting laws", namely (2.130), is used in the proof that a relational ideal J on a proper relation algebra gives rise to a congruence relation (see Th. 111). 12. Axiom of Converse The Axiom of Converse says that for every class A there is a binary relation whose elements are all the ordered pairs obtainable from ordered pairs in A by interchanging left and right sides. VA1BVX(Z
G B O- x is a set A ^V3z{x = (y,z) A {z,y) £ A)).
The uniqueness of the converse of a class follows from the Axiom of Extensionality, so define the converse of A to be the class (2.132)
A-1:={{Vlz):{3;,V)EA}.
Note that A-1 C V2 whenever A-1 exists. With this notation the Axiom of Converse can be shortened to
or simply " 1 exists).
12. AXIOM OP CONVERSE
53
Elementary results involving converse with relative sums and products are collected in the following Th. 17. Several parts of this theorem have been given descriptive names for later reference. Almost all parts of Th. 16 above and Th. 17 below have abstract algebraic counterparts that hold in all relation algebras. Converting between these two settings requires some care, because we are dealing here with classes of sets, rather than classes of ordered pairs. For example, the converse of (2.143) can fail, for if R and 5 contain no ordered pairs, then i?" 1 = S " 1 = 0, but this does not imply R C S, which would indeed be the case if R and 5 consisted exclusively of ordered pairs. T h e o r e m 17. The following statements hold for any classes The converse of a class is a binary relation.
R,S,T,P,Q.
i ? " 1 C V2.
(2.133)
(~1~1) Conversion is an involution on binary relations. (2.134)
OR" 1 )" 1 = -Rn v2>
R'1 = (fln V 2 ) ~ \
Conversion preserves the universal relation and the empty relation. (2.135)
V" 1 = ( V 2 ) " 1 = V 2 ,
(2.136)
0" 1 = 0.
(I" 1 ; t " 1 ) Laws connecting conversion with relative multiplication and relative addition (2.137) (2.138)
CRIST1 (RjSy1
=S~1\R-\ =S~1iR~1.
Conversion distributes over union, intersection, difference, and symmetric difference. (2.139)
{RUSy1 = R~1US~1,
(2.140)
(Rnsy1
= R~1ns~1,
(2.141)
{R^Sy1
= R~1~S~\
(2.142)
(Resy1
=
(
R~1QS~1.
)
Conversion is monotonic.
(2.143)
RCS => R'1
CS~\
Conversion commutes with relational complementation. (2.144)
R'1 = (V 2 ~ J R)" 1 = V 2
2. SET THEORY
Laws of properties (compare (2.113)—(2.115)^) (2 145) (2 146)
(RnV\S)\T = ( Z? 1 1 \ / l G\ 4- T 7 (XL LJ V J 1 1 J-
B|(5 |vnr), flKS-MVUT).
( T h . K) De Morgan's Theorem K [65]. (2 147)
R\SCT
^
(Bno) Forms of Tarski s axiom Rio-
(2 149)
SD^Iiilii"1, RIR-^SC 5D5|i?-!|B,
(2 150)
R-^C
(2 148)
R~X\R\SC
SDWR\R-\
(2 152)
RIR^ISC D R-^iRjS)^ i5C
(2 153)
B|(iFit5)C 5 3 (5tiFi)|ii,
(2 154)
R^\WS)Q 5D(5fii)|iFi, R\(R-^S)c 5D(5fB-1)|B,
(2 151)
(2 155)
-J D
|xt,
—^ ^ C 4|- JXjiJX D\lD— , — -1 l^O
La ws related to Peirce',> Law and transitivity. R^TD
(2 156)
(fltS-^lCSfT),
(2 157)
R\TC
(2 158)
(iit^KSf-^r), R\T C RIS-^SIT, (Ri~S^)\(SiT), ii|TC R\S=?iS\T, (RtS)\(S=?tT), ii|TC RlStS-^T.
(2 159) (2 160) (2 161) (2 162) (2 163)
Formulas for equation-solving. If R,S,r.r e v (2 164) (2 165)
2
, then
> SCR-ijT
« iJCTfS- 1
RCSjT ^ > 5-i|iiCT «
RT-iCS.
Tarski's equivalences. (2 166)
1
husnT-
= 0 ^ TIROS'1 =0 « 5|TnB"
12, AXIOM OP CONVERSE
Cycle law.
=T|S~1niE (2.167)
(rot) Modular laws and rotation. (2.168)
R\SnT=(Rf]
(2.169)
R\Sf\T =
(2.170)
R\SnT =
(2.171)
R~1\SnT =
(2.172)
i?|S"1nT =
n irx|T) n T, n T, n T, n R\T) n T, 1 nr.
Zigzag laws. (2.173) (2.174) (2.175)
1
P
T.
(2.176) Of course, the laws in Th. 16 and Th. 17 can be proved by appealing to the definitions of the operations involved, but many of them can also be given equational proofs that rely on earlier laws. For one of the miscellaneous laws from Th. 16, first note that, by various distributivity laws, we have R\S
n (x f Y) = (R n (x u x))\s n (x f Y) c((Rr\X)\sn(x^Y))u(iznx)|s.
However, rot
c (i?nx)|(5nx x;y1 c(ijnx)|(snf)
~ -mon, |-mon Rio
l-mon so, combining this with the previous equation, we get
c (Rnx)\suR\(Sr\Y).
56
2. SET THEORY
So far we have listed many laws whose abstract relation-algebraic versions are provable from the relation algebra axioms R1-R5 and R7-R10, the ones that do not involve the identity element 1'. These laws could therefore be said to belong to the "calculus of relations without identity". To construct a set-theoretical identity relation we need to upcoming Axiom of the e-Relation. 13. Axiom of the e-Relation The Axiom of the e-Relation asserts the existence of a binary relation consisting of all ordered pairs for which the left side is an element of the right side. 3EVx(x
e E O 3a3b(x
= {a,b)Aae
b)).
A less formal statement of the Axiom of the e-Relation is that {(0,6) : a £ 6} exists. A more formal statement of the Axiom of the e-Relation that does not rely on the Axioms of Extensionality and Unordered Pairs is 3EVc(c£E O 3a3b(a£b A 3y3z(yx(x€c
O I = I / V I= Z ) A
Vz (x e y x = a) A
Vx{x E z O i = a V i = 6)))). The Axiom of the e-Relation implies the existence of a class that is unique by the Axiom of Extensionality. Let E be this class, that is, (2.177)
E:={{a,b) : o £ fo},
and let (2.178)
Id : = ( F r T t E ) n ( E - 1 t E ) ,
(2.179)
Di := E ^ E U E ^ E .
E is the e-relation, Id is the identity relation, and Di is the diversity relation. We expect E, Id, or Di to be proper classes. Reasons for their names and a few simple laws concerning the identity and diversity relations are given in the next theorem. For example, Id is the class of pairs of identical sets, while Di is the class of pairs of distinct sets. Note that Id is the intersection of the superset and subset relations on sets. Theorem 18. (2.180)
E=
(2.181)
Id = {(a, a):ae\/}
(2.184)
= {{{a}} : a € V},
2
(2.182) (2.183)
{(a,b):a£b},
Di = V ~ l d ={{a,b) E ~ T t E = {(a,6> 1
-aCb},
E " t E = {(a,6) : a D &}.
:a^b},
13, AXIOM OP THE e-RELATION
Id and Di partition the universal relation. (2.185)
Id U Di = V2,
(2.186)
Id n Di = 0.
Id and Di are symmetric. (2.187)
I d " 1 = Id,
(2.188)
D i " 1 = Di.
Id is an identity for relative multiplication, and Di is an identity for relative addition. (2.189) (2.190) Subclasses of Id are symmetric relations. (2.191)
R C Id =s- R'1 = R.
(Id-laws) Identity laws. (2.192)
Id cTF^jR,
(2.193)
Id C R-1 f S ,
(2.194)
Id C
(2.195)
Id C
(Di-laws) Diversity laws. (2.196)
R'\R~1 C Di,
^o 1 nT^
D
o— i z^- rv
{z.Lui)
ii\ii
(2.198)
R^IRC Di,
\— Ul,
(2.199) iJ^'llcDi. For any classes A and B, let A^ B = A\B~1. With the addition of the Axiom of the e-Relation (which entails the existence of Id) it becomes possible to replace the Axioms of Relative Product and Converse with a single axiom that guarantees the existence of A f B, namely the sentence 5s in Chapter 2, §3. The converse and relative product can then be obtained as follows. 1
A\B =
=\d\A, 1
1
88
2. SET THEORY
14. Axioms of the calculus of relations In the last few sections, the Axioms of Unordered Pairs, Relative Product, Converse, and the e-Relation have been added to the list of axioms that give rise to the calculus of classes, namely, Extensionality, Empty Set, Intersection, and Complementation. Here is a complete list of the axioms at this point, together with some of their consequences. Extensionality : membership determines identity, Empty set : 0 exists and is a set, Complementation : R exists, Intersection : R n S, R U S, and the universal class V exist, Pairs : the sets {x,y} and (x,y) exist for all sets x and y, Relative product : R\S and the relative sum R'f S exist, Converse : R~x exists, e-relation : the class E exists, so the classes Id and Di exist, The Peirce-Schroder calculus of relations of the nineteenth century is concerned primarily with the empty set, complementation, intersection, union, ordered pairs, relative product, relative sum, converse, the unit relation V2, the identity relation Id, and the diversity relation Di. The Axioms of Extensionality, Unordered Pairs, and the e-Relation are products of the twentieth-century set-theoretical setting. 15. Kinds of relations The class R is said to be symmetric iff R~x C R, i.e., VssVj^a;, y) 6 R~x => {y,x) G R)- A class may be symmetric without being a binary relation. For example, every class that contains no ordered pairs is symmetric. Theorem 19. (i) 0, Id, Di, and V2 are symmetric binary relations. (ii) For every class R, R'1 n R, R'1 U R, R~X\R, RIRT1, R'1 f R, and Rf R~x are symmetric. (iii) If R and S are symmetric classes, then Rtl S and RU S are symmetric classes. The binary relation R~* n R is the largest symmetric relation included in the class R, and it is called the symmetric part of R. If R C V2, then the relation R~x U ii the smallest symmetric relation containing ii, and is called the symmetric relation generated by R. A class R is transitive iff R\R C iZ, and R is a transitive relation if R is transitive and R is also a binary relation. A class may be transitive without being a binary relation; any class that contains no ordered pairs is transitive. We say that a class R is dense iff R C R\R. Since R\R C V2 for every class R, a dense class must be a binary relation. Theorem 20. (i) 0, Id, Di, and V2 are transitive dense relations. (ii) The following statements are equivalent. R is a transitive binary relation,
15. KINDS OP RELATIONS
R\R C R C V3,
RCR-ifR. PROOF. Proof of (ii): The equivalence of the first two conditions holds by definition. For the proof of the equivalence of the other parts, note that the last two conditions imply fiCV2, take i i = S = T in (2.164) and get R\RCRCV2 «- RClf^fR «- RCRfW1. D Peirce [199, 4.94] observed in 1893, Yet really, the form If? is all-important, inasmuch as it is the basis of all quantitative thought. For the relation expressed b j it is transitive. . . . This is not only a transitive relation, but it is one which contains the identity under it. . . . But it is further demonstrable that every transitive relation which includes identity under it is of the form If I. (Peirce [199, 4.94]) We formulate Peirce's theorem as follows. Theorem 21. Let R be any class. (i) RfR-1 is transitive and Id C RfR-1. (ii) The following statements are equivalent: R is a transitive binary relation and Id C R,
P R O O F . Part (i): By (2.194), RfR-1 contains Id. For transitivity, we have (Rf R-^KRf R-1) C RfiR-t^fR 1
C Rf Di fTF T
= RfRr
Peirce's Law, f-mon (2.198), f-mon (2.190), (2.108)
Part (ii): By part (i), if R = Rf R-1 then i i is a transitive binary relation that contains the identity. For the other direction, assume R\R C R C V2 and Id C R, We get R C RfTF1 by Th^20. For the opposite inclusion, from Id C R get V2 n R C Di, hence V2 n R-1 C Di by -1-mon,_(2J.44), and (2.188). Relatively add R on the left of this last inclusion, to get R f R-x = R f(V2 n R~1) C R f Di = R. D Although Peirce's theorem was stated only for RfR-1, it is also true for R- f R, i j " 1 fR, and RfR~1. Here is a corollary that embodies this observation. The list of equivalent statements can be considerably extended. In this connection see Schroder [215, p. 338] and Ng [187, p. 2]. 1
2. SET THEORY
Theorem 22. For any class R, the following statements are equivalent: R\R\J\d CRC V2,
A class R is said to be an equivalence relation if R C V 2 ,ii is transitive, and R is symmetric. Theorem 23.
(i) 0, Id, and\/2 are equivalence relations. (ii) // R and S are equivalence relations, then R(l S is an equivalence relation. The next theorem lists several different ways, drawn from Tarski [227], TarskiGivant [225], and Chin-Tarski [49], to assert in the calculus of relations that a class is an equivalence relation. See Th. 308, where these conditions are all shown to be equivalent in an arbitrary NA. Theorem 24. For every class R, the following statements are equivalent: (i) R is an equivalence relation, (ii) R\RC RCR-1, (iii) R\R = R = R-\ (iv) R\R-X = R, (v) R~1\R = R, (vi) R C V2, RIRT1 £ R, and R'^R C R, (vii) RCV2, R\R C R, and R\R C R. PROOF, (i) => (ii): Assume that R is an equivalence relation. Then, by definition, R is a binary relation which is transitive and symmetric. This means that R C V2, R\R C R, and R~l C R, so we need only note that
R
= R n v2 (2.134) (2.143)
= (R ) c R-1 (ii)=^(iii): Assume R\RC RCR-1. Then R-1 Q(R-
r1 V
=R
2
(2.143) (2.134) (2.133)
Combining this with the assumption that ii C i i " 1 gives i i " 1 = R. Also, X
\R
(2.174)
15. KINDS OF RELATIONS
61
C R\R\R
(2.122), (2.123)
C R\R
(2.122)
From this inclusion, together with the assumption that R\R C R, we conclude that R = R\R. Thus (iii) holds. Obviously (iii) implies (ii), (iv), and (v). Because of (2.133), (iii) also implies (i) and (vi). Therefore (i)-(iii) and equivalent, and each of them implies (iv), (v), and (vi). (iv) => (iii): Assume R\R~X = R. Then R C V2 by (2.103), so ii" 1 = (RlR'1)'1
hyp.
= (ii"1)"1^"1 = R\R~
(2.137)
X
(2.134)
= R
hyp. -1
From this and the hypothesis we get R\R = R\R = R. Thus (iii) holds. By a very similar calculation, (v) implies (iii), so we have shown that (i)-(v) are equivalent. (vi)=>(i): Assume R C V2, R\R-1 C R, and R~*\R C R. The latter two inclusions do not imply that R is a binary relation, which is why (vi) includes an explicit statement to that effect. First we obtain itT 1 C R as follows. R'1 C R-^iR'1)'1^-1
(2.174)
1
(2.134)
= R~ \R\R~
1
C R\R-* CR
(2.122)
From R~ * C R C V2 we derive R = R~ *, as we did in this proof that (i) implies (ii). From this last equation and either the second or the third hypothesis, we conclude that R is transitive. Therefore (i) holds. We now know that (i)-(vi) are equivalent. (iii) => (vii): Assume R\R = R = i?" 1 . Then R C V2 by (2.103), and (2.169) = R\(RnR)
hyp.
= R\®
(2.60)
= 0
(2.111)
so R\R C R~ by (2.21) and (2.47). By a similar calculation we also obtain R~\R C R~. (vii) => (vi): Assume R C V2, R\~R C S, and B|i? C S. Then ^
^
(2.171) (2.123) (2.60) (2.111)
62
2. SET THEORY
so i r 1 R C R, and D— f ( D r~i D1 D \ 1 D — ^ RIR-1 r-,1 JX ^— ^iT 1 1 iX\ILJ -T c (flnfl)lfl"1 = 0 -R"1 =0 X
so R\R~
C R.
(2.172) (2.123) (2.60) (2.1H) D
For a given equivalence relation R and an arbitrary set a; € V, we will define x/R to be the equivalence class of x, that is, the class of sets that are in the relation R to the set x. We do not yet have enough set-theoretical axioms to prove the existence of such a class for every R and x. Even if x/R does exist, it need not be a set. For example, V2 is an equivalence relation under which all sets are equivalent to each other, so the equivalence class of every set is the universal class V, which may not be a set. The existence of the equivalence class x/R follows from the Class Existence Theorem; see (2.390). In case the equivalence relation R is also a set, we can eventually prove that x/R not only exists but is a set. This conclusion is encapsulated in (2.420).
16. Coextensivity Next we define a special operation on couples of classes that yields a "coextensivity" relation. For all classes R and 5, let (2.200)
RoS := (RjS)n(RjlS).
In expressions involving o and |, perform | first, followed by o. When o shows up with fl, U, f, and other binary operations, we tend to use parentheses for clarity. An ordered pair of sets (a;, y) is in the relation RoS iff the .R-image (see p. 94) of the left side x coincides with the S^-image (or S-preimage) of the right side y. A restatement of this characterization and some other basic properties of o are given in the next theorem. The first part shows that o will be obtained from the " in (2.99). definition of f when "V" is replaced by Theorem 25. For all classes E, R, S, and T, (2.201) (2.202)
RoS = {
^
1
^
1
1
and symmetric because C V2,
=R\R~\
68
2. SET THEORY
hence R\R-1 is an equivalence relation; see Th. 24. Theorem 30. If R and S are functional classes, then R\S is a function. If R and S are injective classes, then R\S is an injection. PROOF. If R and 5 are functional, that is, R-1\R C Id and S~1\S C Id, then R\S is a function since R\S C V2 and (2.137), |-assoc R is functional 1
= 5" |5 C Id
S is functional
Theorem 31. If R and S are functional classes, then i?f S is a function. If R and S are injective classes, then R^S is an injection. The associative law for relative multiplication is used in the proof of Th. 30. This usage is essential. The abstract equational version of Th. 30 (see Th. 398) holds in all relation algebras but fails in some semiassociative relation algebra. In contrast, the abstract equational version of Th.31 (see Th. 331) not only holds in all semiassociative relation algebras, but also holds in all weakly associative relation algebras (Maddux [143, Th. 14]). Therefore Th. 31 may be proved without referring to more than three sides of ordered pairs in R and S at any one time, but the same cannot be said for Th. 30: it is necessary to simultaneously consider four objects at some stage of the proof. In the next two theorems we see that if two functions have disjoint domains then their union is a function, while if two functions have the same domain and one is contained in the other, then they are equal. Similarly, the union of two injections with disjoint ranges is an injection; if one injection contains another and has the same range, they are equal. Theorem 32. If R and S are functional and R~1\S = 0, then R U S is functional. If R and S are injective iZ|5 - 1 = 0, then R\J S is injective. A s s u m e R-X\R
PROOF.
C Id, S ' ^ S C Id, a n d R'^S
S-X\R = S'^iR-1)'1
(2.230)
= 0. T h e n
= (R-^Sy1 = 0"1 = 0,
so R U S is functional because = (R-1US-1)\(R{JS) 1
1
1
"'-dist 1
= R~ \RUR~ \SUS~ \R[JS~ \S
|-dist
C Id U 0 U 0 U Id
R, S are functional, (2.230)
18. FUNCTIONAL AND INJECTIVE PARTS
69
Theorem 33. If R and S are functions, RC S, and R\\/ = 5*|V, then R = S. PROOF. Note that S C S|V by Th. 15(ii) R
c s = s n 5|v = S n R\V
hyp.
= Sni?|(ld U Di)
two cases
C(Sni?)U(Sni?|Di)
|-dist
ci?u(S'ns'|Di)
RCS
= RU0
S is functional
= R D
18. Functional and injective parts For every class R let (2.231)
~R:= Rn(R~i\d), —> and refer to R as the functional part of R. For every class R let (2.232)
R:=Rn(\~
and refer to R as the injective part of R. Much of the next theorem may be summarized by saying that the functional part of a class is a function, the injective part is an injection, a class is a function iff it is its own functional part, and an injection iff it is its own injective part. Theorem 34. Let R be an arbitraryclass. (2.233)
RURCRHV*
(2.234)
R = Ro\d,
(2.235)
*R=\doR, —>
(2.238)
iJ
(2.239)
R is a function,
(2.240)
R= R & R is a function.
|.RCId,
2. SET THEORY
The injective
part is an
injection:
(2.241)
two cases
—>
c (2.247)
Hence rot RCR (2.254)
c
(2.231) CR\R
l-mon
For (2.253), apply (2.252) to R, and invoke R = R.
D
T h e o r e m 37. If R is a class, then (2.255) (2.256) (2.257)
R\R
1
(2.258) (2.259)
CId, C Id,
R=
(2.260) (2.261)
R\R= (RH (RH V\R
)^)\R.
In general, relative multiplication does not distribute over intersection, but certain distributive laws do hold when some of the classes involved are functional or injective. T h e o r e m 38. Let R, S, and T be arbitrary classes. Then (2.262)
'
18. FUNCTIONAL AND INJECTIVE PARTS
(2.263)
S\RDT\R=
73
(SnT)\R =
If R is functional then (2.264)
R\(SnT) = R\SDR\T,
(2.265) If R is injective then (SnT)\R = S\RDT\R. PROOF. For part (2.262), calculate as follows. R\SnR\T C R\(SnR
\(R\T))
rot
C R\(Sn\d\T)
-assoc, (2.250)
= R\(Sr\T)
c
R\Sr\R\T
-mon
c
R\SDR\T
-mon
T h e o r e m 39. —¥—¥
(2.266)
r~ /'pi
—y pW
pi D r~ ( pi
D\~^
DID
—y —y
(2.267)
K\K ^ [K\K)
(2.268)
—f
pi p
( pi
4—4—
4—
,
p\~^
K\K — {it\K)
,
(2.269)
p i p (— ( p i p \ ^ ~ iX iX ^— ,
(2.270)
4— 4— p i p (— ( p i D\4— i T i T ^— i
(2.271)
Pi p / Pi p\^~ iX iX —
—
4—
PROOF. Proof of (2.266): Note that R\R C R\R by |-mon since R C R. Also note that from R\RnR\R\D\ = R\R n R\(R\D\)
|-assoc
= R\{RnR\D\)
(2.262)
= R\ , is a functional part and hence is functional. Both pieces are binary relations, and hence both are functions. Finally, these two functions satisfy the condition required in Th. 32 for their union to be a function. Indeed, we have
1(5 C\ R)
|-mon, '-mon
C D i n ( l d f ( S n f l ) ^KSTli?)
Di-law, f" 1 , (2.187)
= Di n Id
(2.153)
19. PROJECTION FUNCTIONS
The next theorem shows that if F be a functional class, then F U (Id ~(.F|V)) is a functional class that is defined everywhere it could be, in the sense that every set is the left side of an ordered pair i n F U (Id ~ F\\l). Theorem 41. If R is a functional class, then R\J (Id ~i?|V) is functional. P R O O F . Let S = R U (Id ~ R\V). Then
S~ \S C R
|(iZU-R|V) 1
~ -mon, |-mon
1
= R~ \RU(R~ \Rjj)
|-dist
C Id U 0
R is functional, Rio
Theorem 42. If R is functional, then R\V n (R\SoT)
C
R\(SoT).
PROOF.
R\Vn(R\SoT) rot |-mon C R\((R~L\R\S^T)
n (R~L\R\S^T))
Peirce's Law, |-assoc
C ii|((ld|5tT) l~l (5fT))
R is functional, |,f-assoc, (2.148)
D 19. Projection functions The main goal of this section is to prove the existence of these two binary relations: l \ \
x
i D l
1^1
x
i U
t
V
J ;
{{{x,y),y):x,yeV}. It will be shown in (2.306), (2.307), and Th. 47 that these binary relations can be obtained from E and Id by using only U, PI, ~ , |, and f. Our starting point is a construction due to Tarski.
2. SET
THEORY
Theorem 43 (Tarski [227, 3.9, p. 108]). Suppose C is any class. Let (2.272)
A :=
(2.273)
B:
Then A and B are injections and the following statements are equivalent: VxVy(xeVAyeV 1
=> 3z(Vw({w,z)
EC O w = xVw = y))),
2
A\B- =V . Tarski's construction, given in (2.272) and (2.273), is mentioned and applied several times in his 1942-43 manuscript Tarski [227, pp. 108, 111, 156, 209, 207, 220-221]. Its converse dual, involving the functional part instead of the injective part, is used by Tarski-Givant [240, 4.6(ii)]. One of Tarski's variations on this construction comes close to our goal, which is Th. 46 below. A class C is called extensional iff (C" 1 o C) fl C " 1 ^ ^ C Id. For example, E is extensional.
Theorem 44 (Tarski [227, p. 207]). Suppose C is a class. Assume (2.272), (2.273), and let (2.274)
Ai : = ^ n V | B n ( V | C t C ) ,
(2.275)
Si :=-BnV|An(V|CtC).
Suppose C is extensional. Then the following statements are equivalent: (x,z)£A1A(y,z)eB1,
(2.276) (2.277)
3u3v(Vw({z,w)
eC^w
= uVw = v)A
yw({u,w)
G C w = x) A
yw((v,w)
£C
)
rot —)
|
(2 .278)
c
(2 .248)
|-normal Consequently two cases
n (c|Di u C|D[) n C|Di) u (
fl-dist, |-dist def. of C the previous equation.
Proof of (2.286): Let R = C\C~CV. First, we have
^
^
^L/
L/ I I u
= 0IC" 1 = 0
I I JX ui j u
1
1
roil,
(2.248) -normal
1
19. PROJECTION FUNCTIONS
From this we get
u cF) n
=cn
two cases
= Cn n-dist, |-dist previous inclusion Finally,
previous inclusion
c idle"
C
is functional
= c-\ Parts of Th. 45 were designed for the proof of the following theorem. are
T h e o r e m 46. For every class C, and all x,y,z £ V, the following equivalent:
(2.287) (2.288)
statements
{x,y)eCVA{x,z)£C*, 3u3v(yw({x,w)eC
w = y \/ w = z). Consider an arbitrary i u £ V. If u> = 2/ or UJ = 2, then (v,ui) £ C by (2.295) or (2.294), respectively. For t h e converse, suppose (2.296)
{v,w)€C.
By (2.294) and (2.296), (2.297)
{x,w)€C\C.
If w = y then we are done, so suppose (2.298)
{y,w)€D\.
Then (x,w) G C^Di by (2.298) and the hypothesis that (x,y) £ C 9 . We have C^IDi C V 2 ~ C ' since Cv is functional, so (a;, to) £ C^. From this and (2.297) we get (2.299) Now (a;,«) G {C^^C^y
(w,a;> € ( C | C ~ O ~ \ by (2.292), so
by (2.299). But (C\C ~ C " ) " 1 | ( C | C ~ C ^ ) ^ C Id by (2.249), so (w,«> £ Id, i.e., w = z, as desired. w = y). Consider an arbitrary w £ V. Next we prove V m ((u,w) £ C First note that if w = y then (u,w) € C by (2.291). For the converse, assume (u,w) £ C. From this assumption, (2.290), and (2.249), we get G C - ' I C C Id, so w = y, as desired.
19. PROJECTION FUNCTIONS
81
Finally, we show that Vw((x, w) 6 C <S> w = uVw = v). Let u i 6 V . If w = u or w = v, then (x,w) G C by (2.291) or (2.294), respectively. For the converse, assume (2.300)
{x,w)eC.
If w = u we are done, so assume (2.301)
<M,w)GDi.
From (2.290), (2.300), and (2.301) we get (x,w) G C l~l C*|Di. ( C n C * | D i ) ^ by (2.293), so
But (x,v) G
(w,v) G ( c r n c " l > | D i ) " 1 | ( c n c r * | D i ) ^ c id, by (2.249). Thus w = v, as desired. Case B. Assume (x,z)eC\C. 1
By (2.283), C\C = (C* n C )^ = C^ n C*, so (2.302)
(1,2)6^
and (x, ^} e (C* n C)\C.
But C^ is functional and (x,y) G C^ by hypothesis, so it follows from (2.302) that y = z and (2.303)
{x,y)e(C*nC)\C.
By (2.303), there is some v G V such that (2.304)
{x,v)eC*,
{x,v)eC,
and
{v,y)eC.
However, we have (x,u) G C* by (2.290), and C* is functional, so (2.304) implies that u = v. Hence (2.305)
and (u,y) E C .
{x,u)€C
Since u = v and y = z, all we need to show is that Vw(((x,w) e C
w = M)A
({u,w) £ C i6 and 0 < i, j < 15, then (2.330) (2.331) (2.332) (2.333) (2.334) (2.335) (2.336) (2.337) (2.338) (2.339)
M(vi = VJ) = {Ri n Rj)|V n W, M(vi e VJ) = (Ri|E n Rj)|v n w, M(-. 3 s ( B is a powerset of ^4)). The uniqueness of powersets follows from the Axiom of Extensionality, so for any set A 6 V, we may define 56 (A) to be the powerset of A: (2.401)
Sb(A):={x:x€V
A x CA}.
The Powerset Axiom is equivalent to
yA(A e v => Sb(A)e V), and is also equivalent to the following equation in the calculus of relations.
Alternatively, we may simply let (2.402)
SJ^tE-'fEJoE.
Then 56 is a functional class by Th. 28, so we may use the functional notation "56 (A)" and regard (2.401) as a theorem instead of a definition. Finally, we may express the Powerset Axiom by this equation.
V2 = Sb\V. Theorem 70. Ifx,yeV
then x\y,x x y, x^1 e V.
PROOF. By Th. 68, x U y is a set since x and y are sets. By the Powerset Axiom, applied twice, 56 (56 (x U y)) is a set. However, xxy C 56 (56 {x U y)), so x x y is a set because every subclass of a set is a set. By the Union Axiom, applied twice, 56 (56 (x U y)) is a set, hence (56 (56 (x U y))) x (56 (56 (x U y))) is also a set by what we have proved so far. However, ( I X J ) U (^ - 1 ) ^ (56 (56 (x U y))) x (56 (56 (x U J/))), so x\y and x~x are sets. 29. Partial orderings, meets, joins, and lattices A class R is domain-reflexive iff R = (Id PI R)\R, and range-reflexive iff R = i?|(ld fl R). By (2.103), every domain-reflexive (and every range-reflexive) class is a binary relation. Furthermore, by (2.191), (2.187), and (2.140), R is domain-reflexive iff R = (Id l~l R'1)^ and range-reflexive iff R = R\(\d n -R"1). R is reflexive iff R is both domain-reflexive and range-reflexive, that is, R = (Id D _R)|.R|(ld D R). An alternative form of this definition, used by Tarski-Givant [225, p. 110], is that R is reflexive iff R D Id n (R\V U V|.R). Note that R is reflexive iff \/x\/y({x, y) € R V {y, x) € R => {x, x) € R).
29. PARTIAL ORDERINGS, MEETS, JOINS, AND LATTICES
99
R is antisymmetric iff R fl iZ" 1 C Id, hence R is antisymmetric iff VxVy({x,y) £ R A {y,x) £ R => x = y). R is asymmetric iff R fl iZ"1 = 0, hence R is asymmetric iff \/x\/y({x,y)
£ R => (y,x) y}>Tl{x>z}} f° ra n x,y,z € B. A lattice ordering of B is complemented if R is bounded and, for every x € B, there is some y such that Y\{xi y} = I I B and $^{x; v) = S ^ - ^ lattice ordering is Boolean if it is distributive and complemented. In a distributive lattice, an element can have at most one complement. The converse was disproved by R. P. Dilworth [68], who showed there are non-distributive lattices in which every element has a unique complement.
30. Axiom of Infinity The Axiom of Infinity is that an infinite set exists. There are many ways to say this. In the version used by Godel [80], the Axiom of Infinity states that there is a nonempty set C such that every element of C is a proper subset of another element of C: 3c{Mx
e C E V ) A Vx(x e C = * 3y(x CyAye
C))).
Godel's axiom of infinity can also be expressed as an equation in the calculus of relations, such as (v|E~V|((E-i|E~E-i|E)|EnE))|E|V. Here is another axiom of infinity, which asserts the existence of a set containing the empty set and closed under the formation of successors: 3o(0 e C e V A V ^ e C ^ i u W e C)).
31. AXIOM OF CHOICE
101
With the help of this axiom we may deduce the existence of the set of natural numbers, (2.411)
u:= {0,1,2,3,4,5,6,...}. 31. Axiom of Choice
The strong form of the Axiom of Choice is that there is universal choice function, i.e., a functional binary relation that picks an element of every nonempty set: 3c (C is functional A Vx(x is a nonempty set 3y(y E x A (x, y) 6 C))). Let C be a class whose existence is guaranteed by the Axiom of Choice. Recall, for comparison, that the Axiom of the e-Relation asserts the existence of the class E = {(x, y) : x € y}. Using E, the conditions satisfied by C can be expressed more succinctly in the calculus of relations. Here is one way: CCE"1,
CnC|Di=0,
C|V = E"1|V.
The Axiom of Choice is equivalent to the assertion that every binary relation contains a function with the same domain. Here are two well-known consequences of the Axiom of Choice. Theorem 72 (The Well-Ordering Theorem, Zermelo [258, 259]). Every set can be well ordered. PROOF. Let s be any set. Prom the Axiom of Choice first deduce the existence of a function g : Sb (s) ~{s} —> s such that g(x) £ s ~ x for every proper subset x C s. To obtain g, just compose complementation relative to s (see (2.317)) with the universal choice function:
(2.412)
g := ( ^ ( ( ( P l i ^ s)}^1)
~ Q ^ " 1 o E) n V3)|C.
Let K C Sb (s) be the intersection of all sets X satisfying the following conditions: x £ X => x U {g(x)} £ X,
Y
a
^-{JY
ex.
K exists by the Class Existence Theorem and the form of its definition by a set-bounded formula: (2.413)
K := {k : Vx(X E V A (V^a; E X => x U {g(x)} E X)
feex)}. Then s = \J K and s is well-ordered by R, the relation that holds between a e s and b G s iff there is some k e K such that a G k and b £ k, that is, (2.414)
R:=
102
2. SET THEORY
Theorem 73 (Zorn's Lemma [261]). If R is a partial ordering of X and every linearly ordered subset of X has an upper R-bound, then there is a maximal element for R. 32. Axiom of Regularity The Axiom of Regularity states that every nonempty class has an element with which it has no common elements: VA(3b(b G A) => 36(6 e i A V ^ ^ V i ^ 6))). This axiom is essentially due to J. von Neumann [253, p. 231, Axiom VI4]; the version used here is due to P. Bernays; see [80, p. 7]. It was proved consistent relative to the other axioms by von Neumann [253]. As Godel [80, p. 6] said, it "is not indispensable, but it simplifies considerably the later work". Using it, we can now prove that the class of all sets is not a set. Theorem 74. For every class X, X 0 X. In particular, V ^ V. PROOF. Suppose X e X. Then X G V, hence {X} £ V. Every element of {X} has an element in common with X, namely X itself, contradicting the Axiom of Regularity.
33. Ordinals and cardinals A class X is an ordinal if every element of X is a subset of X and X is wellordered by E. This definition is due to J. von Neumann [253]. R. M. Robinson [211] noted that, in the definition of ordinal, the condition that X be well-ordered by E can be replaced, owing to the Axiom of Regularity, by the condition that E is linear on X. A class X is a cardinal if X is an ordinal and there is no bijection between X and some element of X. The Axiom of Choice is a major ingredient in the proof that for every set X 6 V there is a unique cardinal \X\, called the cardinality of X, for which there exists a bijection with domain X and range |X|. For more of the elementary theory of ordinals and cardinals, and more complete developments of basic set theory, see, for example, Godel [80], Kelley [122], Mendelson [169], or texts by Vaught [251], Suppes [223], Halmos [88], Monk [179]. We will simply assume elementary results about ordinals, cardinals, real numbers, complex numbers, integers, number theory, and so on. 34. Dedekind-MacNeille completion Dedekind's method of cuts, as generalized to partial orderings by MacNeille [136], is presented here in details using the calculus of relations. There is a precedent for this. Long before MacNeille's work, Schroder [215, §23, pp. 346387] used the calculus of relations to study "Dedekind's Kettentheorie". The Dedekind-MacNeille completion provides an alternative method for constructing the completion of a Boolean algebra; see Th. 215 and the proof of Th. 217.
34. DEDEKIND-MacNeille COMPLETION
103
34.1. Upper and lower bound operators. We start with upper and lower bound operators. For every class R, (2.415)
R1':= (E^1-\ R) o E,
(2.416)
Rl : = (E I T t-R~ 1 )< > E -
By Th. 28, R^ and R^ are functional. An ordered pair of sets (x, y) is in R} if the right side y is the set of upper _R-bounds of the left side x: (2.417)
(x,y) £Rr <S> y = {b : Va(a G x => (a,b) G R)} G V.
If x = 0 then {b : Va(o G x => (a, 6) G R)} = V, and V is not a set by Th. 74, so 0 is never in the domain of R^, even when R is a set. From definitions (2.400) and (2.416) we have f| = El. Recall that the relation Id o E maps each set to its singleton, as is illustrated here: Id oE , Furthermore, if R 6 V then R o E is the function that maps each set a E V to {x : (a,x) G R}. It follows from these observations and (2.421) and (2.422) in the next theorem that
Ro E(a) = R*({a}) = R1({a}) for every set a 6 V. Other parts of the next theorem are improvements on parts of Th. 25 that result from considering E instead of an arbitrary class. Compare, for example, (2.425) with (2.227), (2.426) with (2.223), and (2.427) with (2.225). Theorem 75. Assume that S and T are arbitrary classes and R is a set. f O A ~\ Q\
( E? — \
yZi.^LO)
yTt
)
1
Z?J* — ix
,
( E? — 1 \4yTt
)
T
(2.419)
E- |V = i? |V = ^ | V ,
(2.420)
V2 = (i?oE)|V,
(2.421) (2.422)
RoE = {\doE)\R\ RoE = (Id oE)|i?*,
(2.423)
(i?" 1 )oE = (Id o E ) | ^ ,
(2.424) (2.425) (2.426)
(R
(2.427)
(R-
(2.428) (2.429)
(2.430)
(RoE)\(E-1\SoT) = R\SoT,
(floE)l(E" 1 )*
=R\E~1oE,
(2.431)
( J RoE)|E 3T = B n V 2 ,
(2.432)
(RoE^E'1 = R,
nT — ix
,
104
2. SET THEORY
(RoE)*\f]
(2.433)
= R1',
(R'1 oE)*|P| = Rl,
(2.434) (2.435)
(RoEy\{J
= R*.
R?\R^ and R^\R^ are expanding: (2.436)
R^R1 C E ^ f E ,
(2.437)
R^R* C E ^ f E -
R^ and R^ reverse inclusions: (2.438) (2.439) (2.440) (2.441)
i? t ~ 1 |(E rr tE)|.R T C E ' ^ E , ( E ^ f E) l~l E-1|V C fl^E"1 f E ) ! ^ " 1 , R^^iE^1 (E I I T tE)n(E- 1 |V)
CR^iE'1
R?\R^ and R^\R? are transitive relations, because (2.442)
i? T |i? i |i? T C JRt,
(2.443)
R^R^R1
C R1.
If R is transitive, then (2.444)
((ETTt-R"1)n(E"1|lrT
(2.445)
Ul i Z t = (^ T
PROOF. Proof of (2.419): By (2.417), to prove the inclusion E - 1 |V C i?T|V, we must show for every nonempty set x that jd : Va(a 6 i => {a,b} 6 R)} is also a set. By Th. 60, the class { 6 : V o ( a £ i => (a, b) £ R)} exists for any classes x and R, so let y = {b : Vo(a £ x => (a,b) £ R)}- Our assumption that x is nonempty implies y C R*(x). (This inclusion may fail if a; = 0.) Since R is a set, R*(x) E V by Th. 68. It follows that y is a set since subclasses of sets are sets. For the opposite direction, assume that x is the left side of a pair in .R^V. By Th. 60, {6 : Va(a Ex => (a,b) E R)} is a set, hence, as we observed after (2.417), x cannot be empty, and is therefore the left side of a pair in E ^ V . Proof of (2.420) and (2.421): By (2.223) we have (2.446)
(Id oE)|_RT = (Id o E)|((E T T t R) E) C (Id"f R) o E = Ro E.
Then V2 = (ldoE)|V
(2.245)
C (ldoE)|(Vn(ldoE)-1|V) -i I
= (Id oE)|(E" old)|V
rot
34. DEDEKIND-MacNeille COMPLETION
c (Id oE)|(E
tDi)|V
= (Id oE)|E" 1 = (Id oE)|i? 'IV c (R oE)|V
(2.419) (2.446)
c V2 so we have shown somewhat more than (2.420), in fact, V2 = (ldoE)|B t |V = (i?oE)|V.
(2.447)
Now Id o E, R?, and Ro E are functional by Th. 28, so (Id o E ) ^ is functional by Th. 30, hence (Id o E ) ^ = Ro E by (2.446), (2.447), and Th. 33. Proof of (2.422): By (2.221),
(2.448)
(ldoE)|iT = (ldoE)|(E" 1 |BoE) C Id \Ro E = Ro E.
By Th. 2.245, V2 = (Id o E)|V, but we also have V2 = i T | V by Th. 68 since R is a set, hence V2 = (Id o E)|iT|V. Furthermore, (Id o E)|iT|V C (Ro E)|V C V2 by (2.448), so
(2.449) V2 = (ldoE)|ir|V = CRoE)|V. By Th. 28, Id o E and RoE are functional, while R* is a function by Th. 62, so (ldoE)|iT is functional by Th. 30. Finally, (ldoE)|iT = .RoE by (2.448), (2.449), and Th. 33. Proof of (2.423) and (2.424): By (2.418), (2.421), and (2.422). Proof of (2.425): (-RoE)l(E -LoS) CRoS = (RoS) n(iioE)|v
(2.227) (2.420)
C(iJoE)|((E" 1 ofl- 1 )|(iioS'))
rot, (2.207)
C(floE)|
(2.227)
o5)
Proof of (2.426): r 0R,^S)o j r = ((E:f5)oT)r l(ii oE)|V C (R oEJKCE"1 oR- -^K^t^oT)) C (R oE)|((E~T oT) J
C (R tS)oT
(2.420) rot, (2.207) (2.223) (2.223)
Proof of (2.427): Use (2 .225). Proof of (2.428): Set 5 := E" 1 andT == E in (2.426). Proof of (2.429): R\SoT = (R\i>oT)n(R oE)|V C (Ro
1
C (Ro CR\SoT
1
l
(2.420)
*>R~ )\(R\SoT))
rot, (2.207)
^ oT)I
(2.221) (2.221)
2. SET THEORY
Proof of (2.430): Set S = E" 1 and T = E in (2.429). Proof of (2.431): (i?*E)|ErTClnV2
(2.217)
= Sn(iZoE)|V
(2.420)
1
1
C(RoE)\((E~ oR~ )\R)
rot, (2.207)
C^oE)!!^
(2.216)
Proof of (2.432): (RoE^E'1 CR
(2.214)
= RD(RoE)\\/
(2.420)
1
1
C(RoE)\((E~ oR~ )\R) CfiloEllE"
rot, (2.207)
1
(2.214)
Proof of (2.433): (i?oE)*|P) = (E- 1 |(i?*E)oE)|((E rT tE" 1 )oE)
(2.400)
= (E- 1 |(i?*E)tE- 1 )oE
(2.426)
) oE
duality, f-assoc (2.431)
= R?
(2.415)
Proof of (2.434): By (2.433) and (2.418) Proof of (2.435): (i?oE)*||J = (E" 1 |(floE)oE)|(E" 1 |E" 1 oE) 1
(2.394), (2.399)
1
= E" |(floE)|E" oE
(2.429)
1
= E" |i?oE
(2.432)
= R*
(2.394)
Proof of (2.436): Before giving the proof, note that the hypothesis (x,y) € could be written more conventionally as y = R^(R?(x)), in notation whose use is justified by the functionality of these relations. In particular, (x, R^(R^(x))} € it^l-R^, so, by the proof we are about to give, (a;, R^(R^(x))} € E" 1 f E, which, written more conventionally, becomes x C R^(R^(x)), showing that R?\R^ is expanding.
C(E- 1 t-RtE)|(E-itiZ-ifE) _
= 1^1
E
_
1
(2.200), (2.415), (2.416) ^ (2.138)
(2.162)
34. DEDEKIND-MacNeille COMPLETION
Proof of (2.438): We have
(2 .416) (2 .200), f-assoc (2 .162), f-assoc
C E-i
and (2.415) (2 .207) (2 .200)
(2.160)
C
Proof of (2.439):
rot, Peirce's Law (2.419) rot (2.438) Proof of (2.440): Apply (2.438) to R'1. Proof of (2.442):
(2.436), (2.437), |-assoc
C
(2.200), (2.416) (2.162) (2.415), (2.200) Proof of (2.444): Assume R is transitive. Then
((I^t^- 1 ) n (E-^iF |-mon Peirce's Law, f-mon, (2.137)
2. SET THEORY
R is transitive, '-moil, f-mon
C E-i
and
-mon (2.162) duality
(2.200) (2.416)
Proof of (2.445): Wee have ( E - ^ E - 1 o E)|((E-i tfl)oE)
(2.399), (2.415)
(E-ilE-if^oE
(2.426) duality (2.415)
and x
(E-lrf.ElK^t E- )oE)
(2.400), (2.394)
(E- 1 |BtfE" 1 )E
(2.426)
1
rT
(E- |((E f-R)oE) fE-^oE
(2.415)
E-iKCE-1 f-R)oE)|[""
duality
Ot
(2.431) duality /p— 1 4- D V
(2.415)
34.2. Restricted upper and lower bound operators. Next we define some restricted versions of the upper and lower bound operators. For every class R, (2.450)
R* :=((ErT:iiR)nV\R)oE,
(2.451)
R^ := ((E^f-R^nVli^oE.
34. DEDEKIND-MacNeille COMPLETION
109
Note that R® and R1^ are functional by Th. 28. Since R^ = (R^1)^, we state or prove some things only for R^. An ordered pair of sets (X, Y) is in R^ if the right side Y is the set of those upper _R-bounds of the left side X that are in the range of R. Indeed, for all sets X and Y we have (2.452)
( X , Y)eR1r
^ Y = {b: 3a((a, b) E R), V o (a E X => (a, b) E R)}.
If X = 0 then {6 : 3 a ((a,6> G R), V a (a G X => {a,b} G R)} = {b:3a((a,b)
ER)} = Ra(R).
It follows that if Ra (R) is a proper class then 0 is not in the domain of R^. If R is set then Ra(R) and Do (R) are also sets and (®,Ra(R)) E R*, i.e., ^ ( 0 ) = Ra(R). For similar reasons, i^(0) = Do(R). Let X £ V. If X ~ Do (R) ^ 0 then {6 : 3o((a,6> G R), Va(a G X => (a,b) G i?)} = 0 and (X, 0) G iJ*. So 0 only if X C Do (i?). Theorem 76.. Assume R is a set. Then 1 (2.453) R* =^(R- )*, Ri' = (R-1)\ (2.454)
R*: V -> Sb (Ra (R)),
(2.455)
R*: V -> Sb (Do (R)),
(2.456)
V2 == ^ | V = ^ | V ,
(2.457)
R1t == (E- f-Rf(E U R-^V)) n (E- f iZf E) n (V|i?f E)
(2.458)
R^ == (E- 1 f R-1 f(E u ij|v)) n (E- 1 f R'11E)
(2.459)
1 1 R*' ^(E- f E)|i?^ C E~ fE,
(2.460) (2.461)
tEC^KE^fE)!^"1, R"\l C E-1f(EUi?|V),
(2.462)
R"-\I ^
(2.463)
(2.466)
n (Vli?- 1 1E).
r-_ 2
C E-it(EUi?-!|V), R*\l I^IR^ C^KE^fE), ^IR1'
(2.464) (2.465)
1
1
R*\l X R*\l
Lft
C R^KE-1 f E ) , -ft
^
->Jj- 1 Dft 1 p-li
.ft
.ft , DiT I DIL
PROOF. Proof of (2.454)-(2.456): Ra (R) is a set since R is a set. For an arbitrary set X 6 V let Y = {b : 3a((a,b) E R), Va(a 6 X => (a,b) E R)}. Y exists by Th. 60, and Y is a set because it is a subclass of the set Ra (R), so (X, Y) G R1'. This shows that V2 = _R^|V. The functionality of R?, mentioned above, and the observation that the right side of every pair in _R^ is a subset of Ra (R) are enough to establish that R® : V -> Sb (Ra (R)). The claim about R1^ holds for similar
2. SET THEORY
Proof of (2.457): First, we get (E-1f.R)nV|.RtE = (E-1t-RUV|i?)fE
(2.46)
= E-1t-Rt(EUi?-1|V),
(2.146) (2.450)
!} R)nV\R)oE = ((E- 1 ti?)nv|BtE)n(((E- 1 t-R)nv|i?)fE)
(2.200)
= (E- 1 ti?t(EUB- 1 |V))n(E- 1 t-RtE)n(V|i?tE) Proof of (2.458): Apply (2.457) to i T 1 . Proof of (2.459): We have
(2.119)
(2.457) (2.162), f-assoc. and
c V|(V|i?tE) c V|flfE
|-mon Peirce's Law, |-assoc
so
(2.467)
( E-T t E)|^ CE^f^tE n V|i?fE
We then get (2.459) as follows R« ' K E - i f E 1 C(((E=Tffl) nv|i?)oE)~~ \((E^^R)nV\R)^E)
(2.450), (2.467)
= (E-1o((E~~]R)nv\Rr^KaE-Tf^nvi^tE)
(2.207)
CiE-'Ui^ C E^tE
Tj
tR)nv\R] f1)\(((E^^R)nV\R)^E)
(2.200) (2.160)
It may be worth noting that we did not need to assume that R is a set. That assumption is used in the next proof, however. Proof of (2.460): (2.456) rot, |-assoc (2.459)
34. DEDEKIND-MacNeille COMPLETION
Proof of (2.461): (2.457), (2.458)
= (E" 1 f(i?f E))|((i?f
f-assoc, "Maws (2.162)
Proof of (2.463): First, we have (2.458) "Maws Peirce's Law
C E T
= E~ fi?|V
|-assoc, (2.102), (2.106)
so (2.461) rot J;
rT
rT
= B |((E t(Eu!RlV))n(E t^|V))
inclusion above
Th. 12 f E)
C
, |-mon
Proof of (2.464): Substitute R'1 for R in (2.463). Proof of (2.465): By (2.464), we have C
E).
Furthermore,
(2.174), |-assoc C
(2.463), |-assoc
C
R^ is functional
C R^IR^KE'1 fE)
(2.459)
1
Now R ^ and R® are functional, so R*^ is also functional by Th. 30. Then R1^^ distributes over intersection by (2.264). Combining these observations with the previous two equations, we obtain
C
f E) n
t E)
112
2.
SET THEORY
= iZ^|iZ^|((E^tE)n(E- 1 tE))
(2.264) (2.178)
Let X := iZ^|iZ^. Then we have shown X|X C X , so we also have
x = xnv|x-1 c x n (y n x\x)\x-' = xnx|x|x~ 1
(2.456) rot
CXIXKX-^X- 1 !^- 1 !
rot
X\X C X, -mon X is functional
(— Y1 /1 J\"V— 1 1 \ jV1 \ LA. LA. Y\)
CXiXlId
= x\x as desired. Proof of (2.466): By (2.453) and (2.465). 34.3. Cones. For every a; £ V, (R~1)* ({x}) is called the ii-cone of a;. Note that y € (R'1)* ({%}) (y,x) € R, and R~1oE sends elements to their ii-cones: R-1 oE = {{x^R-1)*
({*})> : x € V}.
If iZ is transitive then, according to (2.220) with E = E, the function iZ^1 o E sends pairs related by R to pairs related by inclusion, that is, for all p , g € V , (R-'o E)(p) C (fl- 1 O E)(q).
{p,q)£R^
Theorem 77. Assume R is a set. Then (2.468)
(
(2.469)
(ld
PROOF.
l
d
1
^
We have
i?|(ld o E)|i^ C (ijf E ) | ^
Peirce's Law
C (BtE)|(E- 1 t-R- 1 t(EUiZ|V)) j
(2.200), (2.458) (2.153)
From these two equations it follows by Th. 12 that i?|(ld o E)|iZ^ C E, which in turn implies (Id o E)|iZ^ C R~x j- E by (2.164). Combining this last equation with
= (DifE)|ii* "fi?
J
tE)
(2.458)
34. DEDEKIND-MacNeille COMPLETION
Ci^fE
t-assoc,(2.153)
yields (2.470)
(ldoE)|#
CR^oE. 2
For the opposite inclusion, recall that V = (Id o E)|i2*|V by (2.245) and (2.456). But then we also have V2 = (Id o E)|.R*|V C (R-1 o E)|V C V 2 , by (2.470), hence (Id o E)|ii*|V = (R'1 o E)|V. We get (Id o E)\R? = R'1 o E by
Th. 33. Applying this result to R~x gives (Id o E)|i^ = Ro E. Theorem 78. Assume ii G V and R is a transitive relation. Then R-1 o E preserves greatest lower R-bounda in the sense that ifu is a greatest lower R-bound ofX ^ 0j then the R-cone ofu is the intersection of the set of R-cones of elements
ofX. PROOF. Assume n is a greatest lower .R-bound of X ^ 0. We can express this assumption in several ways. We could write u = JJ X if R were antisymmetric, for in that case greatest lower bounds are unique, Yl is functional, and the desired conclusion could be expressed as follows:
u = Y,*X => (R-1 oE)(u) =f]((R-1 oET(X)). In the absence of antisymmetry, we are assuming, by (2.409), {x,u) e (E^jR-1)
n ( E - ' l i F i f-R) n E- X |V.
Let C be the ii-cone of u, that is, C = (R'1 o E)(«) or {«, C) e R'1 o E. By (2.444) and (2.434), we have ((E^jR-1)
n (E" 1 !!^t-R^K-R" 1 o E) C R1 = ( i r 1 o E)* P
so {X, C) € (R~x o E)*| f|, which, written differently, asserts that C = E)*(X)), as desired. A more conventional and less computational proof runs as follows. Suppose that w is in the iZ-cone of w. Since w is a lower bound of X, this gives us (w, u), («, a;} € R for every a; € X. But R is transitive, so {w, x) 6 R for every x 6 X, hence w is a lower .R-bound of X. Therefore x is an element of every JJ-cone of every element of X, i.e., u is an element of the intersection of the il-cones of the elements of X. Thus transitivity of R implies
For the other direction, assume w is in the intersection of the .R-cones of elements of X, Then w is in the .R-cone of every element of X, i.e., w is a lower .R-bound of X. But u is a greatest lower .R-bound of X, so {w,u) £ R, hence w is in the .R-cone of u. Thus (This inclusion holds for all relations, without assuming transitivity.)
114
2. SET THEORY
The next theorem shows that R^1 o E converts least upper iZ-bounds into (-R^I-R^)-closures of unions of cones whenever R is transitive and domain-reflexive. When R is also antisymmetric, this can be expressed as follows:
OR"1 o E)(J2RX) = i ^ f l j K f l " 1 o E)* (X)))). Theorem 79. Assume R 6 V and R is a domain-reflexive and transitive relation. If u is a least upper R-bound of X ^ 0, then the R-cone of u is the set of lower bounds of the set of upper bounds of the union of the set of R-cones of elements of X. PROOF. R is transitive and domain-reflexive, so R\R C R and (Id C\R)\R = R. Recall from (2.410) that u is a least upper .R-bound of X iff (X,u) € J where J := ( E ^ f f l ) n (E-^RjR-1)
(2.471)
nE ^ V .
1
Let C be the .R-cone of u, that is, C = (R' o E)(M) or (u, C) £ R'1 o E. Then {X,C)eJ\{R~1oE).
(2.472)
We wish to show that (X,C) G (R'1 o E)*|U|^|fi 4 , which is equivalent to C = (R^lR^dJdR-1 o E)*(X))). But (R-1 o E)*| \J = (R'1)* by (2.435), so we need only show (X,C) £ (R-1y\Rit\R>;>. Let (2.473)
A :=
(E-loR)\,rl\(R-l)*\R1t\Ris-.
We will show that E\A C E and ^4|E—1 C E" 1 . From these two inclusions we conclude that A C E ^ f E and A C E" 1 fE, respectively, hence, by (2.203), AC ( F r T f E ) n ( E - 1 t E ) = Id.
(2.474)
By Th. 68 and (2.456), we have \/2 =
(2.475)
(R-ly\R1t\Rli-\\/,
so J\{R-loE)C(R-ly\R1t\Rii-,
(2.476) because
JIOR-1 oE) = JKR'1 o E) n ( i T Y l ^ l ^ V
(2.475)
C ( J R- 1 )*| J R fr |^|(((i?- 1 )*|^|^)" 1 |J|('R" 1 oE))
rot
l
1t
i
1
= (R- y\R \R ^\A-
(2.473)
C (R-1y\R'"\Rli
(2.474)
It follows from (2.476) and (2.472) that (X,C)£
(R-1)*^^,
as desired. To prove E\A C A, we first establish the inclusion
(2.477)
E^KEHEIJIV) C (i?-i)
34. DEDEKIND-MacNeille COMPLETION
by computing as follows. E^KEfl E|J|V) C E ^ ^ E n EKE^f-R)^) 1
C E~ |(Inii|V) 1
(2.471) (2.153)
1
= E~ |(E l~l (Id D iJ" )^)
R is domain-reflexive
C
C (R-1)*
(2.200), (2.394)
We use (2.477) to show that
J- 1 |(CR- 1 )*n(E- 1 tE)) = 0
(2.478) as follows.
= J-1\((R-1Y = J~1\((R~1)*n(R-1)*)
(2.477)
We then use (2.478) to show
J-'KR-1)" C
(2.479) as follows.
= j-1\((R-1)*n(E-1}E)
u
cj-'ifF'tE) u J~ = J"1|(ErTtE) We use (2.479) in the final calculation. E\A = EKE"1 oR^J^KR-1)*^^ l
l
1
^
(2.478) (2.473) (2.432) (2.479) (2.457) (2.162), f-assoc
C i?|(i?f i F 1 ! E ) | ( E ^ t ^ t E ) | f l ^ ^
(2.471) (2.156)
2. S E T THEORY
c (i?|i?fE)| n RV c ( i J t E ) ! ^ n R\V
Peirce's Law
C (i?tE)|(E-it-R- 1 t(EU J R|V)) n R\V
(2.458)
C (EUi?|V) n R\V
(2.153)
t-, -mon, R istransitive
C E
Th. 12 X
J
:
To show A\E~ C E~ we first notice that J ~ C i ? " 1 ^ since j c (riffljnE-'lv C E-'KVnEKE^fi?))
rot
C E^^i?
(2.153)
C V\R 1
It follows that l /~ |( J R~ 1 )*|^ fr C J - J | V C iT^VIV = fl-^V. Furthermore, we also notice that J
1V
(2..473)
)
C (RC
X
"t^KE" \R- \E) 1 1 (R~ t^lE" !-R-'fE 1
(2..471), (2.394), (2.200) Peirce's Law, |-assoc
1
c R- Ifl-'tE c R-1 tE
(2..152) R\
I, f-mon
These last two observations give us the first step in the next computation.
R-^y^nR'^v
(2.457) (2.153)
CE Use this last inclusion in the second step: AlE'1 = ( E " 1 * ^ ) ^ " 1 ! ^ " 1 ) * ! ^ ! ^ ^ " 1
C (E'^^IEI^IE"
(2.473)
1
"tfl"1)!^)^"1
(2.458)
1
(2.152)
-1
I-mon
ffl" ) \R )
(2.153) C E"
1
This completes the proof.
(2.152) D
34. DEDEKIND-MacNeille COMPLETION
117
For a more conventional proof in case R is a partial ordering, assume u is a least upper iZ-bound of X / 0. This entails that u is in the iZ-cone of every upper iZ-bound of X, so u is a lower iZ-bound of the set of upper iZ-bounds of X, that is, u £ (R('\Ril')(X). By the transitivity and range-reflexivity of R, a set is an upper .R-bound of the union of the iZ-cones of elements of X iff it is an upper .R-bound of X itself:
What we are trying to prove therefore reduces to (R-loE)(u)=R*(R1t(X)). Suppose w is in the iZ-cone of u, that is, w £ (iZ"1 o E)(it). Then (w,u) £ R. By transitivity of R, this gives us w £ iZ^(iZ^(X))). For the other direction, suppose w is R to every upper bound of X. One of those upper bounds is u, hence (w, u) £ R, i. e., w £ (iZ~1 o E) (u). Recall (p. 76) that a class R is extensional iff (iZ"1 o R) n iZ'^VliZ C Id, and E is extensional. Theorem 80. Let R CM2. (i) // iZ is transitive and extensional, then R is antisymmetric. (ii) // R is reflexive and antisymmetric, then R is extensional. (iii) If R is a partial ordering, then R is extensional. PROOF.
Proof of (i): First note that
iZ"1 n R = iz^lid n id\R c iz-^vn v|iz = .R-^viiz by |-mon and (2.115). Hence R is antisymmetric because iZ"1 niZC (iZ" 1 ffl)n(!R r T t^)riiZ" 1 |V|iZ 1
1
= (iZ" oiZ)niZ" |V|iZ C Id
R is transitive, Th. 20 (2.200) R is extensional
Proof of (ii): Since R is reflexive, we have R = iZK-R"1 n Id) and hence also R-1 = (iZnld)|iZ- 1 , so R~X\V\R = (iznid^iz-'lviiZKiZ" 1 n id) c ( ^ n id)|V|(JR-1 nid). Then, using various parts of Th. 16, Th. 17, and Th. 18, we obtain (iZ- 1 oiZ)niZ" 1 |V|iZ
c (R-1 oiZ) n (id n fl)|v|(id n R'1) = (id niz)|(iz"1oiz)|(id niZ"1) c R\(1FI jR)\\d nidKiz^tiZ)!^" 1 cizniZ" 1 C Id so R is extensional.
R is antisymmetric,
118
2. SET THEORY
Proof of (iii): A partial ordering is reflexive and antisymmetric, and so is also extensional by part (ii). 34.4. Completion of a relation. The next theorem generalizes Dedekind's method of cuts. For partial orderings it is due to MacNeille [136]; see McKenzie-McNulty-Taylor [167, p. 51,53]. We prove it for a binary relation R that is transitive, domain-reflexive, and extensional. By Th. 80(i), such a relation is antisymmetric. If R is also range-reflexive then, since R is already domainreflexive, R is reflexive and hence, because it is antisymmetric and transitive, R is a partial ordering. For an example of a transitive, domain-reflexive, extensional relation R that is not a partial ordering, choose distinct a, 6, c e V and let R={{a,a),{b,b),(a,c),{b,c)}.
Theorem 81 (Dedekind-MacNeille completion). Assume R is a set and a transitive, domain-reflexive, extensional binary relation. Let F=(R-1oE)DR-1\\/.
(2.480) Then
(i) F : Fd(R) -> Sb (Fd(R)) ~{0}, F is infective, and x G F(x) for every xeFd(R). (ii) Every element in the range of F is (R^\R^)-closed, in the sense that F\R1t\R11' = F, in other words, (R1t\Ris-)(F(x)) = F(x) for every x G Fd (R). (iii) F embeds R into the inclusion relation on (R^\R^)-closed subsets, i.e., F'^RIF C E ^ f E; if{x,y) {V). PROOF. Let F be the subset of A consisting of those elements of SI for which the conclusion is true, that is, F = {a:a€A,3v(VCX,\V\
1
1
1
( K I J I ) ) ^
1
|-assoc
C ^~ |(P|P~ n QIQ" )!^
or and r are functional (2.313)
C Id
i9 is functional. D
The clone generated by {i, X) = / % , z) =s- w = y A x = z.
Then 21 is absolutely freely generated by X.
3. GENERAL ALGEBRA
PROOF. Let 03 = {B, (]', v', t'} be any algebra of group type and let / : X — B. Then / C X x B, so / is a subset of the universe of the direct product 21 x 03 and generates a subalgebra h in that product, i.e., h = 6cj /S'(6,&') = d. Assume that (c, d) € ft and /3(a, a') = c. We wish to prove ff (b, b') = d. Now {c, d} cannot be a generator, since otherwise fl(a,a') = e E X, contrary to (3.55), and (c, d) cannot be {i, i'), for this would contradict (3.56) by requiring i = (3(a, a'). It also follows from (3.56) that {c, d) cannot be in the range of the unary operation of 21 x 9$. Therefore (c, d) is in the image of h under the binary operation of 21 x !8, say (c,d) = ((3(v,x),l3'(w,y)) for some (v,w), {x,y} € ft. Then f}{a,a') = c = /?(«,»), so a = v and a' = a; by (3.58). By applying (3.61) to the pairs («, w), (a;,!/) £ ft we get b = it? and &' = y. These two equations imply /3'(&, b') = {3'(w,y) but we already have d = fi'{w,y), so d = fi'(b,b'), as desired. This completes the proof of (3.61) for both inductive cases. Since h is functional, contains / , is a subalgebra of SI x 93, and has domain A by (3.59), it follows that ft is a homomorphism from 21 into 03. There are several ways to construct absolutely freely generated algebras; see Henkin-Monk-Tarski [93, pp. 130-1]. We will use the following particular method adapted from Henkin-Monk-Tarski [93, 0.4.19(i)]. For every ordinal a, let (3.63) Fra := f]{U : a C U,%Vy(m,y G U => {0,0}, (1, (x)} , (2, (m,y)) £ V)}, and define the algebra $ta of group type by (3.64)
&a:={Fra,0,v,t)
where* = (0,0), v(x) = (1, (a;)), andp(x,y) = (2, (a;,j/}} for alia;,?/ g Fra. Notice that i and every element in the range of either v and j3 is an ordered pair, but no ordinal is an ordered pair, so Ra09) n a = Ra(t))na
= { ( } n a = 0.
The unicity property (2.95) of ordered pairs implies that # t a satisfies the sufficient conditions in Th. 91 for an algebra to be absolutely free. We therefore have the following theorem. Theorem 92. ^ r a is absolutely freely generated by a. The next theorem asserts the existence of algebras of group type that axe absolutely freely generated by an arbitrary set. Similar theorems hold for all other types of algebras. Theorem 93. For eneri/ nonempty set X £ V there exists an algebra of group type that is absolutely freely generated by X.
136
3. GENERAL ALGEBRA
PROOF. Choose an ordinal a for which there is a bijection A : a —> X. Let Y = {X} x (Fra ~ a) and A = X UY. Note that X and Y are disjoint, a consequence of the Axiom of Regularity, for if x £ X fl Y then there is some w £ Fra~a such that x = (X,w) £ X, but then X€\£\£X. This contradicts a consequence of the Axiom of Regularity, that there can be no 6-loops. Next, imitate the structure of $ta on A by treating elements of X as if they were in a, and elements of Fra ~ a as if they were in Y. Specifically, for all x, x £ X let /3(x,x') = (X, (2, (A(a;), A(a;')>» and let v(x) = (X, (1, A(a;)>). Let t := (X, (0, 0)). Then 21 = (A,/3, v,t) is the desired algebra. In view of Th. 90, we refer to $va as the absolutely freely a-generated algebra of group type. This algebra exists (is nonempty) even when a = 0, since every algebra of group type has a constant (an "operation of rank 0"). This applies as well to monoids and algebras of relational type, but does not apply to unary algebras, groupoids, or algebras of Boolean type. (Here one may choose to allow algebras with empty universes.) The elements of Fra are called terms, but if a < w and we wish to emphasize that the set of variables is restricted to the first a of them, we will refer to terms in Fra as a-variable terms. The ordinals in UJ are, in this context, called variables. Note that
Fra = | J Frp,
Pairs of terms are called equations, and pairs of a-variable terms are called avariable equations. An equation e = (to,ti) is valid in (also true in) an algebra 21 = (^4, /3, a, L) of group type if for every function a : n —> A called, in this context, an assignment, the unique homomorphism a' : Frn —> A that extends the assignment a has the property that o'(io) = a'{ti). Write 21 |= e if e is true in 21. Consequently, 21 |= e h(t0) = ft(ii)for every ft 6 Hom{Fr^,iA). For every term t £ Fra, define free(t) to be the smallest subset of a that generates a subalgebra containing t, and refer to free(t) as the set of free variables of t. By Th.82, ||free(t)| < to. Theorem 94. Let n < u. Consider an algebra 55 = {B,/3',v', if) of group Let E be the type. Let Gn he the n-ary clone generated by smallest relation such that - GS, - (k, P/b n Bn+1) € E for every k < n, and - ifto,ti 6 Frn, TO,TI E Gn, and {to, TO) , (ii,ri) 6 E, then G E and (P'(to,U), (TO]?-1 n TiIQ"1)|/3'> £ E. Then (i) E is a function that maps the terms in Frn onto the elements of the n-ary clone Gn: E:Frn^ Gn,
10, FREE ALGEBRAS
137
(ii) An equation {to,ti} is valid in the algebra 33 iff the terms in the equation determine the same element of the clone Gn, that is,
»H*o,*i> iff E(t0) = Efa). (iii) For every assignment a : n —¥ B and every term to € Frn, if a' is the unique homomorphism extending a, then a (to) = E(to)(a(n - 1),... ,a(0)). If 0 ^ n < w, then for any n-ary term to € Frn in the absolutely freely ^-generated algebra #r n of group type, and any algebra 25 of group type, we will write £(f in place of E(ta), where E is the evaluation function described in Th. 94. Thus, for example, the equation (*o,ti) is valid in 25 iff i(f = *? Suppose we have a set of equations S C Fr^ and a class if of algebras of group type. We will write 9$ |= £ to mean that every equation in S is valid in 33 and K |= £ to mean that 2$ |= £ for every 2$ € if. Let Grp be the class of algebras of group type. A class if C Grp is said to be an equational class if there is an equational axiomatization for K, that is, a set £ C (.Frw)2 of u-variable equations such that K = {33 : 33 1= S, © £ Grp}. if is a finitely based equational class if there is a finite equational axiomatization for K. We say that K C Grp is a variety if K is closed under the formation of subalgebras, homomorphic images, and isomorphic copies of direct products, that is, K = SK = HK = PK. Birkhoff's HSP-Theorem is next. For more about this and related theorems, consult Henkin-Monk-Tarski [93, §0.4] or McKenzie-McNulty-Taylor [167]. Theorem 95 (Birkhoff [31, 228]). Suppose K is a class of similar algebras. The following statements are equivalent: (i) K is an equational class, (ii) K is a variety. PROOF, (i) =^ (ii): Assume if is an equational class. Then there is some set of equations E such that K (= S and, for any class L C Grp,
(3.65) L 1= S =}- L C K. It is straightforward to prove, from K \= S, that HK \= E, SK \= E, and PK 1= S. These last three statements imply, by (3.65), that HK C if, SK C if, and Pif C if. Since the opposite inclusions always hold, if is a variety. (ii) => (i): Assume if = Sif = Hif = Pif. Let E be the set of equations that are true in every algebra in if, i.e., E := {e : e 6 {Fraf, K |= e} Suppose that 21 G Grp is a algebra such that 21 |= S. By Th. 93, there is an algebra $t^ 6 Grp that is absolutely freely generated by A. Let I be the set of all congruence relations on ^r^ with the property that the correlated quotient algebra is in if. (3.66)
/ := {C : C G CodvA, g t A / C G if}.
138
3. GENERAL ALGEBRA
The direct product, over / , of all these quotient algebras is, by its construction, in PK, but PK = K, so
V : = I I ^AIC e K.
(3.67)
Gel
Next we single out the subalgebra of this direct product generated by those elements that are correlated in a natural way with the elements of A by a function f : A —¥ P. In particular, if a e A then the element of the product correlated with a is the function :CeI). /„ := (a/C Let 6 := &Qm({fa :a€A}) = S 0 ( w ( i i a ( / ) ) . Note that 6 G S{*P} C SK = K. Since $vA is absolutely freely generated by A, the function / from A to elements of %5 extends to a homomorphism h from $tA to 6 . Note that h maps $tA onto Sb (Ei) where Ei is a set and an equivalence relation. Define a function h : A —> Sb {[JieI Ei) by setting, for each a € A, h{a):=\Jpi{*)iei
It is easj to construct small examples which show that ft may not be a representation of 21. However, by the following theorem, it will be a representation if the fields of the relations Ei are disjoint. Theorem 104. Assume E is an I-indexed system of nonempty equivalence relations, that is, 0 ^ I g V, E : I —¥ V, and, for all i,j 6 I , i) n Fd{Ej) = 0 Then (Jiei -®« *
s an
equivalence relation and
13. CLOSURE OP RRA UNDER SUBALGEBRAS AND PRODUCTS
149
via the isomorphism ((RnEi-.iei):ReSb
(\Ji€IEt)).
Furthermore, if SI is a nontrivial algebra of relational type, p : I —> V, and, for all i € /, pi is a homomorphism from 21 into 6b (Ei), then (\JieI pi{a) : a 6 A) is a homomorphism from 21 into &b (Uiej Ei). Th. 104 is useful because any system of representations can be "isomorphically altered" by various set-theoretical devices so that the disjointness condition is satisfied. One case where this procedure is not needed occurs in the next theorem, by which an equivalence relation algebra may always be regarded, up to isomorphism, as the direct product of square relation algebras. Theorem 105 (Lyndon [133], Maddux [139, Th.8(8)(9)], [142, 1.7(5)]). // E £ V is an nonempty equivalence relation then U£Fd(E)/E
via the isomorphism ((RClU2 :Ue Fd (E)/E) :ReSb
(E)) .
Furthermore, for each U e Fd(E)/E the function (RnU2 : Re Sb (E)) is a homomorphism from &b (E) onto 9ie(U). Theorem 106. \{&b (E) : E'1^
= E £ V} = P{91e (U) : U € V}.
Let E'1^ = E e V. If 0 ^ E, use Th. 105 to conclude that 6b (E) e P{9k (U) : 0 ^ U e V}. If 0 = E, then 6b (0) = *Re (0) € P{9k (U) : U E V}. For the opposite inclusion, note that the direct product of a system of algebras in {$Re (U) : U € V} is isomorphic to another such product from which all factors of the form *He (0) have been deleted. The second product is in turn isomorphic to an equivalence relation algebra by Th. 104 and the remark after it. The key point here is that nonempty equivalence relations do not have any empty equivalence classes. PROOF.
Theorem 107. RRA = SP{*Re (U) :U eV} = PRRA. PROOF.
First, note that RRA = IS{6b (E) :E~1\E =--Ee V} = SI{6b (E) :E~1\E =--E€ V} = SP{me(U) .UeV}
(3.78) (3.44) Th. 106,
and then PRRA = PSP{«Re(I7) : U
ev}
CSPP{me(U) :U € V } = SP{mt{U) -.u e V} = RRA
(3.46) (3.45)
150
3. GENERAL ALGEBRA
C PRRA.
(3.45)
The primary goal of the next few sections is to prove the theorem of Tarski [232] that HRRA = RRA. For this we will need basic facts about homomorphisms, kernels, and congruence relations of representable relation algebras. After that we present some results concerning special kinds of relations in proper relation algebras. From the previous sections we see that it is fairly easy to show that RRA is closed under the formation of subalgebras and direct products of systems of algebras. It is, however,somewhat harder to show that RRA is closed under homomorphisms, that is, RRA contains all it quotient algebras. The first proof that HRRA = RRA is due to Tarski [232] in 1955. Combined with SRRA = PRRA = RRA, this is enough to conclude, on the basis of BirkhofPs Th. 95, that RRA has an equational axiomatization. In 1964 Monk [176] proved that no equational (and, in fact, no first-order) axiomatization of RRA is finite. Monk's proof utilizes algebras that arise from projective geometries according to the 1961 construction of Lyndon [135], From the results of Lyndon [135], which connect projective lines and planes to relation algebras and their representations, respectively, together with the 1949 theorem of Bruck-Ryser [45] on the nonexistence of projective planes, it follows that there are infinitely many projective geometries, each consisting of a single line with a finite number of points, whose corresponding algebra of relational type is not in RRA. Furthermore, it is quite straightforward to show that any nonprincipal ultraproduct of these algebras is in RRA. It follows from this observation by general model-theoretic considerations that RRA does not have any finite first-order axiomatization, let alone an equational one. In 1965 Monk [177] converted the algebras of relational type derived from projective lines into cylindric algebras, and, by essentially the same argument involving ultraproducts, showed that there is no finite equational (or even firstorder) axiomatization for the class RCA3 of representable 3-dimensional cylindric algebra. In 1969 Monk [180] went on to prove that there is no finite equational (or first-order) axiomatization for RCAa whenever 3 < a < u. To deal with the dimensions beyond 3 he invented new cylindric algebras whose nonrepresentability is based on Ramsey's Theorem; see, e.g., Graham-Rothschild-Spencer [85]. These cylindric algebras, converted back into algebras of relational type, provide another proof that RRA has no finite equational axiomatization. Indeed, they are the ones used in the proof of Monk's Th. 458 below. 14. Relational ideals Recall that an algebra is in RRA iff it is isomorphic to a subalgebra of &b (E) for some equivalence relation E. Therefore, to study homomorphisms on representable relation algebras, it suffices to consider homomorphisms on subalgebras of equivalence relation algebras. Let us suppose that E is an arbitrary equivalence
14. RELATIONAL IDEALS
homomorphism h
congruence relation h\h-L
relational ideal {R : h(R) = ft(0)}
( C o E ) n ( i x V)
C
0/C
{{R,S):ReSeI}
I
TABLE 2. Homomorphisms, congruences, ideals
relation, and that 21 is a subalgebra of ©6 (E). Note that E is actually determined by 21 since E is the Boolean unit of 21, that is, 1st is an equivalence relation, but we choose to call it E (a more suggestive notation). We will define the notion of relational ideal and prove theorems which say, in aggregate, that homomorphisms, congruence relations, and relational ideals are interchangeable in the sense that each determines one of the others by some particular construction, and that these constructions are inverses of each other. Table 2 contains a summary. The first line asserts that an arbitrary homomorphism h 6 HoSl on a proper relation algebra St C SHe (E) gives rise (according to Th. 85) to the congruence relation h\h-1 £ Co%, which appears in the second column of the table, and h also produces (according to Th. 110 below) the relational ideal {R ; h(R) = fc(0)}, called the kernel of h. In the second line, an arbitrary congruence relation C E CoSt on the algebra 21 gives rise to the relational ideal 0/C and (according to Th. 84) to the homomorphism (C o E) n (A x V). Finally, an arbitrary relational ideal I on SI gives rise (according to Th. I l l below) to a congruence relation {{R, S) : R,S G A,RQ S G I}. The homomorphism corresponding to the relational ideal I is : R,S
S e I}oE) n(A xV).
From every homomorphism h we get a congruence relation ftlft"1 by Th. 85. Furthermore, every congruence relation arises in this way, i.e., every congruence relation C is equal to ft|fe~x for some homomorphism h by Th. 84. Thus the function ^ft|fe-1 : ft € i?o2l) maps Ha3l onto CoSt. Conversely, every congruence relation C 6 Co% gives us a homomorphism, in fact, a homomorphism from the algebra 21 onto the quotient algebra 21/C. However, not every homomorphism arises this way. The ones that do are homomorphisms that we may call quotient homomorphisms —homomorphisms onto quotient algebras. Every homomorphism corresponds to a quotient homomorphism by Th. 85. The image of 21 under a homomorphism ft is isomorphic to the quotient of SI with respect to the congruence relation h\h~x. Thus we may write
Since not every homomorphism arises from a congruence relation, the fact that the processes leading from one to the other are inverses of each other must therefore be expressed in terms of congruence relations. Specifically, we see from Th. 84 and Th. 85 that every congruence relation C is equal to the congruence
152
3. GENERAL ALGEBRA
relation ((Co E) n (A x V ) ) | ( ( C o E ) ( l ( i x V))- 1 , which is determined by the homomorphism (CoE)n(AxV), which is in turn determined by the congruence relation C. We say that I is a relational ideal of the proper relation algebra St C 6 b (E) if I has the following properties: (i) ICA, (ii) if Re A and RCS el then Re I, (iii) if ii, S 6 I then RUS el, (iv) if ii 6 I then E\R g i, ii|B € i, and ii" 1 g I. A relational ideal I of 21 is proper if I ^ A. I is a maximal relational ideal of SI if I is a relational ideal of St and I = J whenever J is a relational ideal of SI such that / C J. A couple of useful observations are made in the next theorem. Theorem 108. Suppose I is a relational ideal of a proper relation algebra SI and E = 1^. For every R e A, R e I & E\R\E el. I is proper iff E {Res VS(SG A => (Res
<S>
(0, S) £ C)),
el R 6 7. But Re R = 0 and 0 6 7, so R 6 7. For the converse, suppose R £ I. To show R £ ker(h) it suffices to prove, for any S £ A, that fleSe/» 5 £ /. I f S e / then R U S £ I by closure of / under unions, but Re S C RU S, so Re S E I, and, conversely, ii Re S E I then 5 ' C i ? U 5 ' = i?U(i?eS') G / b y (2.82) and closure of 7 under unions, so S £1. 15. S and H commute on proper relation algebras Theorem 112. Assume E £ V is an equivalence relation. Then (3.99)
HS{6b (E)} = SH{6b (E)}.
PROOF. We may assume E is not empty, since &b (0) is a 1-element algebra of relational type, so S{6b (0)} = {6b (0)} and HS{6b (0)} = H{6b (0)}. But H{6b (0)} is the class of 1-element algebras of relational type and this class is closed under subalgebras, so SH{6b (0)} = H{6b (0)} = HS{6b (0)}. We have HS{6b (E)} D SH{6b (E)} by (3.43), so assume 35 <E HS{6b (£)}. Then there is a subalgebra 21 C 6 b (E) and a homomorphism h E Horn (01, 23) from 21 onto 93. Let
(3.100) J :={R:RCE,RC\£ker(h)}. Next we show that J is a relational ideal of 6 b (E). Suppose R,Se,J. Then there are R', S" such that R C R' £ ker(h) and S C 5" £ ker(h). We have RUS C R'US' and R' U S' £ ker(h) since ker(h) is closed under unions, so R U S £ J. Thus J is closed under unions. For closure under subrelations, suppose R C S £ J. Then S C S' £ ker(h) for some 5", but then R C S' £ ker(h), hence R £ J. Finally, for closure under conversion and relative multiplication by E, suppose R £ J. Then R C S 6 ker(h) for some S. Since ker(h) is a relational ideal on 21, we have S^1 £ ker(h), S\E £ ker(h), and E\S £ ker(h), but by monotonicity we also have R'1 C S~\ R\E C S\E, and £|7Z C E\S, so i?" 1 G J, ij|£ G J, and
E\Re J. Let C := {{R, S) :R,SCE,ReS
/ : = (C o E) n (Sb (E) x V),
e,7],
IB. S AND H COMMUTE ON PROPER RELATION ALGEBRAS
We then know that C G Co&b (E) by Th. I l l , / is a homomorphism from 6b (E) onto 6b (E) /C by Th. 84, J = ker(/) by Th. I l l , and C = / I / " 1 by Th. llO(iii). To complete the proof we will show that g is an isomorphism from 23 into 6b (E)/C, for then 23 G IS{6b (E) /C) C ISH{6b (E)} = SIH{6b (E)} = SH{6b (E)}. Here is the situation: 6b (E)
1/ 23 — 9 —^ 6b (E) /C To show g is functional, first note that /i|ft-1 C f\f~1, for if {R, S) € h\h-1, then fl,SeA and h(R) = h(S), so Re S e ker(h) C J, hence (i?, S) G C = / I / " 1 . Then, since / is functional, we have
g-'b = r ' l W O l / c r ' K / i r 1 ) ! / = (/- L I/)I(/- 1 I/) c id|id = id. To show g is injective, assume (b,c) G g\g-1 Then (&,c) G ^ ~ 1 | ( / | / ~ 1 ) | ^ ! so there are ii, S € A such that 6 = ft(B), (R, S) € / I / " 1 , and h(S) = c. Then (fl, S) € C since / I / " 1 = C so fle5 G J. By (3.100), ReS C|e ker{h), but fcer(/i) is closed under subrelations and i? 9 S £ A, so i? 9 5 6 ker(h), hence /i(i?) = /i(S'), i.e., 6 = c. We only show that g has the homomorphism property with respect to the operation +sg- Computations for the other operations of 23 are similar. Let 6, c € B. Since ft is a homomorphism onto 23 we may choose R, S € A such that h(R) = b and h(S) = c. Then (6, R) G /i" 1 and (c, 5) eh'1, while (fl,/(fl)) G / and (S,f(S)) G / , so (b,f(R)) G ft"1!/ = 5 and {c,f(S)) G ^ J | / = S) *-e-) sC7) = f(R) a n d S(c) = /(-S*)- The operation corresponding to +sg in the quotient algebra 6b (E) /C is Lt/c- What we wish to show is gib + c) = g(b) \J/Q g(c). Since h and / are homomorphisms, we have h{R US) = h{R) + h{S) =b + c, i.e., {b + c,RUS} G ft" , and 1
f(RUS) = f(R)U/cf(S), i.e., {R U 5, /(i?) U/ c /(5)) e / , hence , the universal quantifier V, a (possibly empty) set C, whose elements are called constants , a (possibly empty) set J-, whose elements are called function symbols, a (possibly empty) set 1Z, whose elements are called relation symbols, a function rank : T U 72. U {=} —> OJ ~{0} that assigns a positive integer (a rank) to each relation symbol, a positive integer to each function symbol, and the positive integer 2 to the equality symbol, that is, rank(=) = 2, o a predicate equality symbol equality symbol = whose intended interpretation is equality between binary relations, o predicate operation symbols +, ~, ;, and ". The items marked with are standard for any language C, while the ones marked with o are the additional features of Tarski's extended languages, obtained by adding operators on binary predicates and a new type of atomic formula that expresses equality between binary predicates. A nice language C is one in which all the sets and elements above, and several others sets constructed below (terms, formulas, etc.), are appropriately distinct and disjoint. This means, among other things, that the following sets are pairwise disjoint: V, {=}, {-.}, { ^ } , {V}, C, T, R, rank, {=}, {+}, {"}, {;}, and {"}. It also means that these sets appropriately contain or are entirely disjoint from the sets of terms and formulas of £ that are defined later. The variables, equality symbols, connectives, universal quantifier, and predicate operators are
168
4. LOGIC WITH EQUALITY
regarded as fixed and consequently the same for all languages. In particular, we chose UJ as the set of variables to be used for equational theories, so it would be natural to include the stipulation that v = UJ1 as part of the definition of nice language. With respect to the fixed set V U {=,->, => ,V, = , +,~, ; , " } , any particular choice of sets C, J-, and 72, and rank function rank will either produce a nice language or it will not. We will usually consider only nice languages. For any given nice language £, let Sym(£) be the set of nonlogical symbols of £, that is, (4.1)
Sym(£) :=CUfU1l.
We regard the language £ as determined by its nonlogical symbols and its rank function. We will express this dependence by writing £ = £(C, J-, 72., rank), leaving the fixed symbols unmentioned. By the way, the rank of a constant in C, if it were defined, would be 0. Assume that £ = C(C,r, 72., rank) is a nice language. Our next task is to construct sets of terms and formulas. 1.2. Terms. The set Tm(£) of terms is the smallest set that contains VUC and is closed under the following rule for forming terms: if / £ T is a function symbol of rank r = rank(/) £ w ~ 1 and ti,... ,tr are terms, then the ordered pair (/, ( t i , . . . , tr)) is a term, usually written fti tr, or / ( i i , . . . , tr), or f(ti -tr), or even ii/ Tm(£) andv : Tm(£) -> Tm(£) by /3(t,t') = (b,(t,t')) and v(t) = (b,(t)) for allt,t' € Tm(£). Then (Tm(£),/?,«) is an algebra of Boolean type that is absolutely freely generated byVUC. If C = {i} then (Tm(£),/3, v, L) is an algebra of group type that is absolutely freely generated by V'. The algebra (Tm(£),/3,u,t) is the t e r m algebra of £. In this example the structure of £ was chosen so that the term algebra is a free algebra of group type on countably many generators. The term algebra for an arbitrary language could have infinitely many operations of arbitrarily large finite rank. 1.3. Predicates. The set II(£) of predicates of £ is the smallest set that contains the equality symbol 1', contains every relation symbol R 6 72. that has rank 2, and is closed under the following rule for forming new predicates from old. If A, B are predicates, then the ordered pairs (+, (A, B)}, {~, {A}}, (;, {A, B)), and (", (A)) are also predicates, usually written A + B, A, A;B, and A, respectively. We usually also write II instead of II(£) when £ is clear from context.
1. SYNTAX
169
Theorem 123. For every language C = C(C, J-, 1Z, rank), II has the properties (4.2)-(4.4). For a nice language, II also has the property (4.5). (4.2)
1' <E n ,
(4.3)
{R : R G 1Z, rank(iZ) = 2} C n ,
(4.4)
if A,B €TI then A + B, A, A;B, A €11,
(4.5)
^3 = (II, +,~, ;, ", 1') is an absolutely free algebra of relational type that is absolutely freely generated by {R : R € 7Z, rank(i?) = 2}.
The algebra ?P = (II, +,~, ;, ", 1') is the predicate algebra of C. Since it is an algebra of relational type, we introduce the standard additional algebraic notation from (3.3) and (3.4), letting 1 = 1' + F , 0 = 1' + F , and, for any A,B G II, A B = A + B and A]B = ~A-^B. Although 1' is the same as = , we will use whichever notation is most appropriate for the context. Note that |II| = u even when 1Z = 0, since II always contains the predicates that can be built up from the equality symbol 1' using the predicate operators. Although the rank function rank is not actually defined on every element of II, it does produce the value 2 when applied to any R e II n H, so we extend the concept of rank to all the predicates in II by asserting that all predicates have rank 2, and may even occasionally express this by writing rank(iZ) = 2 whenever R € II. Even for nice languages, 1Z and II may not be disjoint, since both sets contain the relation symbols of rank 2. However, if there are no relation symbols of rank 2 in 1Z, then, for nice languages, 1Z and II are disjoint. 1.4. Atomic formulas. If R € 1Z U {=}, R has rank r = rank(iZ) > 0, and i i , . . . ,tr € Tm(£), then the ordered pair {R, (ti,.. . ,tr)) is called an atomic f o r m u l a . I t i su s u a l l y w r i t t e n R t \ - t r , o rR ( t \ , . . . , t r ) , o rR ( t \ -tr), o r t\Rt2 in case r = 2. Let AtFm(£) be the set of atomic formulas. AtFm(£) :={R(t1,...,tr)
:
R£1ZU{=}, r = rank(iZ), ti,...,tr
€ Tm(£)}.
Let AtFm + (£) be the larger set, called the extended atomic formulas, obtained by augmenting AtFm(£) with formulas obtained by treating the elements of II as binary relation symbols, together with a new kind of variable-free atomic formula, namely, equations between predicates. If A, B e II, then ( = , {A, B)) is an atomic formula, usually written A = B, called a predicate equation . Predicate equations will be used as the sentences of the formalism Cx (see Table 1), so we define Sent x (£) by (4.6)
Sentx (£) := {A = B : A, B € II},
and define the set of extended atomic formulas thus: AtFm + (£) := AtFm(£) U {t1At2 : A G II, , (ip,ip)), (-i, ip), and (V, (x, ip)) are formulas, usually written as ip => ip, -up, and V ^ , respectively. Fm(£) : = f]{X : AtFm(£) C X, if 1JJ,
(4.10)
(pAi/i
= ->(tp =
(4.11)
{T,F}. Theorem 124. Let £ = C(C,J-,TZ, rank) be a nice language. (i) Fm + (£) is the closure of St under => and ->, i.e., (4.14)
Fm+(£) = P){X : St C X, if ip,tp £ X then ~^
and -i, but £ is the intersection of all such sets, so £ is included in all such sets, including Fm + (£). This shows S C Fm + (£). For the opposite inclusion, note first that AtFm + (£) C St C £, next, that S is closed under => and -i, and, finally, if ip € £ and x €V, then \/xip e St C £. Since Fm + (£) is the intersection of all sets that contain AtFm + (£) and are closed under => , -i, and Va, for all x 6 V, and S is such a set, it follows that Fm + (£) C S. Since Fm + (£) is closed under =^, we will use "=>" to denote not only the connective => but also the binary operation /3 on Fm + (£) defined by /3(ip,i/>) = (
ip) = , ( , -i) is an algebra of Boolean type because Fm + (£) is closed under -i and => . Prom part (i) we know that S generates (Fm + (£), =>,-i). Furthermore, from the form of the definitions of formulas in Fm + (£) and the assumption that £ is nice it follows that ip => ip f. St and ->ip ^ St for all {T, F} there is a unique function (4.15)
v : Fm+(£) -» {T, F}, +
such that, for all ip, ip £ Fm (£), (4.16)
v(tp) = v( (
Vvhip)[s] for every loose interpretation M and every sequence s GUM, i.e., either M \/= y vji {tp => tj)) [a] or M \fc)[a]t
(4.60)
M | = tfi[s].
Let de M. Erom (4.59) and (4.57) we get (4.61)
M\=
i/> [a(k/d)].
Prom (4.61) and (4.55) we conclude that (4.62)
either M £
). It follows by Th. 132 and (4.60) that (4.63)
M\=V[a(k/d)].
By (4.62) and (4.63), we get M \= i/f [a(k/d)]. We proved this for every d 6 M, so it follows by (4.57) that M 1= Vv&^ [«]. The next two theorems are lemmas for the proof of Th. 137. The first one states the connection between the substitution of a term for a variable and the alteration of a sequence. Theorem 135. Suppose £ is a nice language, M is a loose interpretation for £ with domain M, a GWM, t,t' G Tm(£), and keu. Then (4.64)
s(k/8M(t))M(t').
sM (Subr*(O) =
PROOF. Use induction on terms. First consider the case when t' is a variable, say t' = v r o . Then, by the relevant definitions, 3 M (v ro ) = sm
if m / k
j
11 m — K
(t;
and \9m [aM(t)
iim^k ifm = ,
so (4.64) holds when t' is a variable. In case t' is a constant, say t' = c € C, we have
184
4. LOGIC WITH EQUALITY
A s s u m e t h e t h e o r e m holds for t e r m s i i , . . . ,tr s y m b o l of r a n k r. T h e n M
(/ti
tT)) = SM(f
G T m ( £ ) and / 6 f ,
SubVth ( t l )
a function
Sllb^ (ir))
= fM(s(k/sM(t))M(t1),..
.,s(k/sM(t))M(U))
s(k/sM(t))M(ft1---tr).
=
D T h e o r e m 136. Suppose £ is a nice language, M is a loose interpretation for £ with domain M, s G UM, t G Tm(£), ip G Fm+ (£), k G u, and t is free for Vk in ifi. Then
M\= G Fm+(£) and for all s G UM, M
iff Af|=Subrfc(V)M}-
We will show that P has the closure properties listed in the definition on p. 175 of "is free to be substituted for", which implies that ip G P, as desired. First we show that if vfe g free(V>), then
(4.66)
ipeP.
Let s € " M . Then the following statements are equivalent: M \= tp [s]
note (a) below
fe
M \= Sub^ (ip) [s]
note (b) below
M
Note (a): s(k/s (t)) and s agree everywhere except possibly at k, but vj, is not agree on the free variables of ip. free in ip by assumption, so s and s(k/sM(t)) Apply Th. 132. Note (b): By Th. 129, Sub*" (ip) = ip. This completes the proof of (4.66). Next we show that every atomic formula is in P. Consider an atomic formula A = B with A,B e Tl. Then A = B € P by (4.66). Next consider an atomic £ Tm(£). Let formula Rti T where R € U U II, r = rank(iZ), and ti,...,tr s G uM. Then the following statements are equivalent: M
\=Rti---tr[s(k/sM(t))], ,
8(k/8M(t))M(tr))
G -RM, i?^,
Th. 135
2. SEMANTICS
Thus every atomic formula is in P. Next we will show P is closed under = and universal quantification with respect to variables that are not free in t. Suppose «/>,£ G P. Let s GWM. Then we have M\=1>[s(k/sM(t))]
iff
M\=SUbvt-(1>)[s],
M\=S[8(k/sM(t))] iff A e)M, M ^ Sub"fe ('4>) [s] or M \= Sub") [sO'/d)] for all d G M,
note (d) below note (e) below,
>f ^ V v . Sub^(V>)H, M^Sub^(Vv^)[s],
(4.57) (4.33)
Note (c): s{k/sM(t))(j/d) = s(j/d){k/sM{t)) since k ^ j . Note (d): Since M vj f var(t), we have s {t) = s(J/d)M(t) by Th. 131. Note (e): We can apply the assumption that tp G P, with s(j/d) in place of s. This completes the proof that P is closed under universal quantification with respect to variables that are not free in t. Since has the required closure properties, we conclude that ip G P, so (4.65) holds. Theorem 137. Let C be a nice language, t G Tm(£), k G u>, andip G Fm + (£). If t is free for Vk in tp, then \= Vvj,^ => SubJh (ip).
186
4. LOGIC WITH EQUALITY
PROOF. Let M be a loose interpretation for £ with domain M, and let s £ M. We wish to show that M \= VVfev? =^ Sub^fc(i^) [s]. For this purpose, assume .M |= V v ^ M - : t follows from this by (4.57) that M \= ip[s(k/sM(t))]. We may apply Th. 136 since t is free for vjt in cp by assumption, so M. \= SubJ'*' ( Sub?(ip) that are not logically valid. For example, \i R £ 1Z, and rank(iZ) = 2 then => Siiby(3yRxy) => 3ySuby{Rxy) = Vx3yRxy => 3w.RSub* = Vx3yRxy => 3yRyy. However, \/x3yRxy => 3yRyy is not logically valid, since if M = {a, b}, a ^ b, and RM = {(a,b) ,(b,a)}, so that RM is simply the diversity relation on the two-element set M, then M \= Mx3yRxy (in M, everything is different from something), but M ^= 3yRyy (in M it is not the case that there is something different from itself). 3. Axiomatization and formalisms A ("syntactical" or "uninterpreted") formalism (see Tarski-Givant [240, §1.6]) is a pair (S, h), where S G V is a set (whose elements are called "sentences" or "formulas") and h is a binary relation (called "provability" and read "proves") which relates subsets of E to elements of S, that is, h C Sb (S) x S, and which satisfies the following conditions for all $ , $ C S and all ip,ip £ S. (4.67) (4.68) (4.69) (4.70)
if f £ * then $ h 0)1.
4. FORMALISMS OP "Barskl-QIvant
(AV)
[V,(
tf) => (V,
V.tf)],
(AVI)
[V.
Vx
(FV)
-.-.p => p,
(FVI)
p => -.-.p.
,
It will be shown below that the axiom schemata (HI) and (HII) are enough to prove the Deduction Theorem for propositional calculus. They can, in turn, be derived from it, so they are, in this sense, equivalent to the Deduction Theorem. In the proof, (FI) can be used in place of (HI), hence (FI) and (HII) are also enough to prove the Deduction Theorem. Since (FI) can be derived via the Deduction Theorem, provability using (HI), (HII), and MP is the same as provability using (FI), (HII) and MP. Lukasiewicz proved that (FIII) is redundant (see (4.85) below) and (FIV)(FVI), can be replaced by (HIII). Axiom schema (HIII) is a law of contraposition, and is one of many axiom schemata that can be used here, such as this variation on (HIII): (HIII')
(-.p => -.V) => ((-"P => ip) => ], where x £ hee(ip).
(AVI*)
However, Monk [178] proved that (AVI) cannot be eliminated entirely. On the other hand, by Tarski [235, Th.3] and the ensuing remark, (AVI) may be deleted entirely if the provision "where x ^ y" is omitted from (AVIII), but this change is not desirable for languages with only a finite number of variables; see TarskiGivant [240, p. 8]. The rules of inference in Tarski's second system S2 are MP and Gen, and its axiom schemata are Lukasiewicz's (LI)-(LIII) together with five others:
(civ)
vx(
v) => WX
v^v),
(CV)
V ^ => V,
(CVI)
(fi => V^v?, where x £ var(ip),
(CVII)
3x(x=y),
where x ^ y,
194
(CVIII)
4. LOGIC WITH EQUALITY
x=y => {
V), where (p is atomic, x € var( V1, M ewen/ v? 6 Sent+(£).
PROOF. Proof of (4.76): Let tp £ Fm+(£). The following five formulas are also in Fm+(£). ip! := (tp => (tp => tp)) => ((ip => ((ip = > ? ) ) = > 95 from 0 using axioms H^ and rule MP, since {rpi,i/>2, ips., ipi, V's} ^ FmJ(£), ^1 is an instance of (HI), tp2 is an instance of (HII), {ipi,tp2,ip3) G MP, ^4 is an instance of (HII), and (ip3,ip4,ips) G MP. Proof of (4.77): Assume ip, ip 6 FmJ(£) and $ C FmJ(£). For one direction, we simply note that if $ \-d tp => ip then $ , tp \-d tp => V by (4.70), and $, (^ \-d ip by (4.67), so *,
ip}. We prove by induction that if v5 by (4.76), so ip G fi. If V1 G $ U Ed then (^> => (^ => ip),ip,
ip from $ using H<j and MP, so $ h d tp => ip and f e d . So far we have shown $ U H d U {^} C Q. To show O, is closed under MP, suppose ip,tp => f £ O, i.e., $ h d y> => ^/) and $ h d 95 =>(?/;=> f). Hence there is a proof (<TI, . . . , o>) of 7J => ^ from $ using H^ and MP, and there is a proof (TI, . . . , TS) of ip => (ip => f) from using H^ and MP. Let Xi := (tp => ip) => ((
£)) => (
(^ => 0) =^ (V =^ 0 . X3 := V => £ an
Then xi is instance of (HI), (xi:°"r,X2) G MP, and {X2,TS,XZ) G MP, so the following sequence of formulas is a proof of tp => £ from $ using H^ and MP: (4.79)
(-
Th. 142 still holds if we replace (HI) with (FI) and alter the proof at this point by setting Xi := (
(V1 => 0) => ((
VO => (
0 ) , X2 : = (tp => ip) => (tp => | ) ,
X3 := V => C.
196
4. LOGIC WITH
EQUALITY
and noting that %i is an instance of (FI), (xi, T S , X2) £ MP, and (x2, 0>, X3) £ MP, so (4.79) is again a proof of ip => £ from $ using H<j and MP. Since $ U 5 j U { ip}. Then $ U Sg U {ip} C 0 and fi is closed under MP, as shown by the proof of (4.77). We need only show 0 is also closed under Gen. Assume ip E il, i.e., $ h 9 ip => ip, so there is a proof (CTI, . . . , ar) of tp => ip from =^ V") =^ (V3 =^ Vj;V)
T h e n (o>,Ci) £ Gen, £ 2 is a n instance of (HIV) since x £ free(y) = 0, a n d (C2,Ci:Cs) ^ M P , so (CTI, . . . , OV,CI>C2J &} ls a proof of ip => ^txtp from $ using H s , MP, a n d Gen. D
7. Implicational fragment Through this section we assume £ = C(C,J-,1Z, rank) is a nice language, n < UJ, and \-d is provability in FmJ(£) using axioms (HI), (HII), and rule MP. T h e o r e m 143. Let tp, $,£ e Fm+(£). Then (4.80)
\-d ip => y>,
(4.81)
h d v? ^> (V> => tp),
(4.82)
h d 95 ^> ((y> =} ip) =} V;),
(4.83)
h ((95 => tp) => ip) => ip,
(4.84)
h
(4.85)
h
(7J => ( ^ => £)) =^ ((^j => ^ ) => ( ^ => ^ ) ) , (ip => (ip => £))
(ip
(ip => ^ ) ) ,
d
(4.86)
h (v? ^> V) => ((V => (^ => £)) => (V =^ 0 ) ,
(4.87)
h
ip => ( ( y =^ ^1) => ((95 => (ip => 0)
=^ f ) ) ;
d
(4.88)
h (v? => V) => ( W => £) => (V => C))>
(4.89)
\-d (ip =>£)=>
(4.90)
\~ ip => ((tp => (ip => £)) => (ip => (,)).
(('P => tp) => (>p => V1)))
Every part of this theorem is very easy to prove with the help of the Deduction Th. 142 and the properties of the provability relation \-d, without any reference to the axioms (HI) and (HII). For example, although (4.80) is a repetition of Th. 142, we now can put forward the following alternative proof. We have tp \-d tp by (4.67). By the Deduction Theorem, treated as an additional postulated property of the provability relation \-d, we conclude that \-d tp => tp. Similarly, (4.82) may be
7. IMPLIOATIONAL FRAGMENT
197
proved by noting first that (ip => ip, tp, ip) is a proof of ip from {ip, tp => ip} using no axioms and rule MP, hence tp,tp => ip \-d ip. We therefore conclude by two applications of the Deduction Theorem that \-d tp => ((tp => ip) => ip). For a similar proof of (4.83), first note that (tp => tp) => ip, tp => p Hd ^ since ((y> => y>) => $, y> => tp, ip) is a proof of ip from {(y> =J> tp) => $, y> =s> tp} using no axioms and rule MP. By the Deduction Theorem we obtain tp => tp \-d ((ip => ?/>) => ^. From this and (4.80) we then obtain (4.83) by (4.68). All parts of Th. 143 may be proved in this way, including (4.81) and (4.86), hence every instance of (HI) and (HII) has such a proof. Thus, by defining \-d in terms of the axioms (HI) and (HII) and the rule M P we have obtained a provability relation that satisfies (4.67)-(4.70) and the Deduction Theorem. Conversely, if a provability relation satisfies (4.67)-(4.70) and the Deduction Theorem, then all instances of (HI) and (HII) are provable from the empty set. It is in this sense that we may say that the Deduction Theorem is equivalent to provability using (HI), (HII), and MP. All these remarks remain true if we refer to (FI) instead of (HI). Let us say that a formula is an arrow tautology if it is a tautology that belongs to the closure of St under => alone. Then Th. 143 contains a selection of arrow tautologies. By the way, Th. 143 asserts the provability of many formulas that are not arrow tautologies according to this definition, but which nonetheless can be seen to be tautologies on the basis of what might be called their "arrow structure", the way they are obtained from simpler formulas using only =>-. In view of Th. 143 we might reasonably ask whether \-d tp for every arrow tautology tp. A short survey reveals that the answer is "no". Suppose we have distinct formulas tp, tp £ St and {£i, £2, £3} C {tp, ip}. Among all of the formulas - (
£0 => (£2 => £a)» - ((*> => £1) => £2) => £s, - tp => (£1 => (£2 => £3)), - (
(£1 => £2)) => £3, - V => ((£1 => £2) => £3), there are 20 tautologies, and all but one of them can be proved using only the Deduction Theorem and properties of provability. The one that cannot be proved in this way is ((ip => ip) => tp) => tp, which is known as Peirce's tautology. Theorem 144. It is not the case that \-d ((ip => ip) => tp) => tp for every tpeSt. PROOF. Use an alternative definition of —> as an operation on a set with more than two elements. Let M be third object distinct from T and F. Define —* as in Table 3. Let C be the => -closure of St. Then, as in Th. 125, every valuation v : St —> {T, M, F} has a unique extension to a map v : C —> {T, M, F}. A formula ip 6 C is an arrow tautology with respect to this new definition of —> if v(tp) = T for every valuation v. It may be checked that every instance of (HI) and (HII) in C is a tautology with respect to this new —>, and the property of being a tautology with respect to this new definition is preserved by MP. But Peirce's tautology is
4. LOGIC WITH EQUALITY
—>
T M F
T T T T
M M T T
F F F T
TABLE 3. Unique 3-element table for independence of Peirce's tautology
not a tautology in this new sense, for if v(tp) = M and v(ip) = F then v(((
ip) => F ) - > M ) - > M
= M.
On the 3-element set {T, M, F}, with T as designated element, there is, up to isomorphism, only one definition of —> that can be used in this way to show that Peirce's tautology is not hd-provable. If we wish to define —> instead on the 4-element set {1,2,3,4} and use 1 as designated element (so that ip £ C is a tautology iff v{ {1,2,3,4}), then there are (counting isomorphic copies) exactly 78 definitions of —> that work, but only 17 isomorphism types. They are shown in Table 4, with the first one presented in two forms, and the other sixteen in abbreviated Since Peirce's tautology is an arrow tautology that is not provable using (HI), (HII), and MP, we may ask whether we will obtain a complete axiomatization of the arrow tautologies if we add all instances of Peirce's tautology to our axiom set. It follows from some results of Tarski and Bernays that this is the case. First we define some additional schemata (consisting entirely of arrow tautologies in the more general sense described briefly above). (Pe) (Ta)
0
Tarski proved that ip is an arrow tautology iff it has a proof using axioms (HII), (LI), (Ta) and the rule MP. To express this more briefly we will say that (HII), (LI), (Ta) are complete for arrow tautologies. . Bernays pointed out that (Ta) may be replaced by Peirce's tautologies (Pe) in Tarski's theorem, so (HII), (LI), (Pe) are also complete for arrow tautologies. Independence of both systems was proved by Lukasiewicz. See Tarski [237, Th. 29 and subsequent remarks]. Furthermore, Lukasiewicz [242, 244] (see also Tarski [237, Th. 30, fn. f]) proved that the shortest single axiom schema that axiomatizies the implicational propositional calculus is (LIV)
f) => (X
7. IMPLICATIONAL FRAGMENT
1 1 1 1 1
—>
1 2 3 4
2 2 1 1 1
4 3 3 1 1
3 3 3 1 1
1 1 1 1
2 3 3' 1 3 3 1 1 1 1 1 1
2 1 1 1
4 3 1 1
3 3 1 1
1 1 1 1
2 1 1 1
3 3 1 1
4 3 1 1
1 1 1 1
2 1 1 1
4 3 1 1
4 3 1 1
1 1 1 1
2 1 1 1
4 4 1 1
3" 3 1 1_
1
2 1 1 1
3 4 1 1
4" 3 1 1_
1
2 1 1 1
3 3 1 1
3" 4 1 1_
1 1 1 1
2 1 1 1
3 3 1 1
4" 4 1
1 1 1
2 1 1 1
2 1 1 1
4" 4 4
1 1 1
3 1 1 1
2 1 1 1
4" 4 4
1 1 1 1
2 1 1 1
3 3 1 1
4" 4 4 1
2 1 2 1
3 3 1 1
4" 4 4 1
2 1 2 2
3 1 1 1
4" 4 4 1
1 1 1
1 1
1—i
M
1 1
1 1 1
1—i
1 1 1 1
TABLE 4. All 4-element tables for independence of
1 1 1 1 1 1
2 3 3' 1 4 3 1 1 1 1 1 1 2 4 3' 1 3 4 1 1 1 1 1 1 2 3 4' 1 1 1 4 1 1 1 4 1 1 1 1 2 3 4' 1 1 3 4 1 4 1 1 Peirce's tautology
For an outline of the proof of Lukasiewicz's theorem, see Church [50, 18.4]. Church [50, 18.3] also has suggestions for proving the theorems of Tarski and TarskiBernays. By (4.88), every instance of (LI) is provable using (HI), (HII), and MP, so the theorems of Tarski and Bernays still hold if we replace (LI) by (HI). Consequently (HII), (HI), (Ta) are complete for arrow tautologies, and so are (HII), (HI), (Pe). So far, we have found four sets of schemata that are complete for arrow tautologies. Each of them consists of three schemata among (HII), (HI), (Pe), (LI), (Ta), and each of them includes (HII). The inclusion of (HII) is essential in the sense that the remaining four schemata are not complete for arrow tautologies. Table 5 shows an alternative definition of —> on {1, 2, 3, 4} with respect to which, with 1 as the distinguished element, the property of being a tautology is preserved by MP and all instances of (HI), (Pe), (LI), (Ta) are tautologies, but some instances of (HII) are not tautologies, since 2—> (3 —> 2) = 2 —» 4 = 4 ^ 1. There are two other choices of three schemata among (HII), (HI), (Pe), (LI), (Ta) that may produce sets of axioms that are complete for arrow tautologies, namely (HII), (HI), (LI), and (HII), (LI), (Ta). In view of (4.88) and Th. 144, we see that the first of these possibilities is not complete for arrow tautologies. To show that (HII), (LI), (Ta) are also not complete, use the definition of —> in Table 6, with
4. LOGIC WITH EQUALITY
—>•
1 2 3 4
1 1 1 4 1
2 2 1 4 2
3 2 1 1 2
4 4 4 4 1
TABLE 5. A table for the independence of (HII)
-
>
•
2 3 4 2 2 4 1 1 1 4 3 3 1 i—l i—l
1 2 3 4
1 1 1 1 1
TABLE 6. A table for the incompleteness of (HII), (LI), (Ta)
(HII) (HI) (Pe) (LI) (Ta) || + - - + - 1 + - + - - 1 + - + - 1 + + - - 1 + - - + 1 - + - + 1
1234 1222 1222 1222 1234 1224 1224
2 2 2 2 2 2
1234 1112 1111 1111 1114 1114 1114
3 3 3 3 3 3
1234 1112 1114 1111 4114 2224 1144
4 4 4 4 4 4
1234 1111 1131 1221 1221 1221 1221
TABLE 7. Incompleteness of some axiom schemata for implication
which instances of (HI) fail since (3 -> (2 -> 4)) -> ((3 -> 2) -> (3 -> 4)) = 4. Table 7 proves that no set of axioms formed by choosing two schemata from (HII), (HI), (Pe), (LI), (Ta) is complete for arrow tautologies. This includes independence for the axiom sets in the theorems of Tarski and Bernays. In the table, "+" indicates that all instances are tautologies with respect to the table given to the right and "—" indicates otherwise. 8. Completeness of (HI), (HII), (HIII') In this section, assume C = C(C, T, 1Z, rank) is a nice language, n < u>, and \-p is provability in Fm+(£) using axioms (HI), (HII), (HIII') and rule MP. By the Soundness Th. 141, whatever can be proved is a tautology. For completeness we must show the converse, that every tautology in Fm^(C) can be proved from 0 using axiom schemata (HI), (HII), (HIII') and rule MP. Theorem 145. For all ip,ip 6 Fm^(£), (4.91)
-«p, ij), —Of => Ip \~P Ip.
(4.100)
(4.91): a proof of J/1 from y>, -195 is
PROOF
1. -, ( -T0 =>
5.
-iip
=> (
(HII)
y)
1, 3, MP 2, 4, MP
= > -MP
6.
P
7.
-nip)
8.
ip) :
=>. ( ( ^ v => v ) => V)
(HIII') 5, 7, MP 6, 8, MP
9. (4.92): By (4.91) and Th. 142. (4.93): a proof of ip from -1^3 => -iip, -up => ip is 1.
-i -
2.
=> ((-1^3 => I/)) => y>)
(HIII') 1, 2, MP
3. h
4. -.p => i 5.
(4. .67) 1, Th. 142
-l^ip
Tl 1.142, (4.70)
-1^3
P
-1^3
=>
5. - . - . v ? h
"
, -«p => -«p h ip
p
(4.95): By (4.94) and Th. 142.
(4. .93) 2, 3, 4, (4.68)
4. LOGIC WITH EQUALITY
(4. 96): 1. 93 h
(4.95), (4.70)
2. 93 h p 3.
—,—,—,
—1—1"-199 = >
4. 93h
(4.70), Th. 142 —K£>, —1—1—iy? =>- y?
h
199.
p
(4.93)
p
1, 2, 3, (4.68)
(4. 97): By (4.96) and Th. 142. (4. 98): 1. 99,-1^,-1-1(99 =^ tp) \-p ip ^
tp
(4.94), (4.70)
P
2. 99, -.t/), -.-.(99 =^ V) !" V P
(4-67)
1
3. 99,99 =^ V !~ V
("y5 => V>, V) !~P V1
1, 2, 3 , (4.68)
p
5. V) => V>
4, T h . 142
6. 99,-1^,-1-1(99 =^ -J/I) h p -it/i
(4.67)
p
7. 99,1V) h -.-.(99 => ip) => -up
6, T h . 142 p
8. -1-1(99 => tp) => -np,-i-i(
tp) => tp \- -1(99 =3- tp)
(4.93)
p
9. V)
5, 7, 8, (4.68)
(4.99): \~p p
1. 99 => tp,^p
( 4 . 9 4 ) , (4.70)
p
2. 99 => V , - ' - ' ¥ ' l" 99 ^> V>
(4.67)
P
3 . 99, 99 = ^ V !~ V> 4. p ^
(p ^
p
tp, -.-.99 h V
1, 2, 3, (4.68)
5. 99 => tp hp -.-.99 => tp 6. -*p,ip
4, T h . 142
=> ip hp -.-.99 => tp
7. -.V>, 99 => tp h
p
tp, -.V> p
8. -1-199 => -,^,-.-.99 =^> tp \- -mp p
9. -nip, p => tp \- -mp
5, (4.70) (4.67), T h . 142 (4.93) 6, 7, 8, (4.68)
10. 99 => tp hp -nip => -199
9, T h . 142
(4.100): 1. 99 ^
tp,^ip
=> tp \-p -up => -.-.99 p
(4.99), (4.70)
2. 99 => tp, -nip => tp \- -«p => -nip
(4.99), (4.70)
3. -up =^ -.-.99,-iV> => -199 h p V>
(4.93)
4. 199 ^> V, ¥> => V1 I" P «/>
1, 2, 3 , (4.68)
D
8. COMPLETENESS OF (HI), (HII), (Hill')
For every valuation v : St —> {T, F} and every formula
VO", st"(v3) h p v?".
PROOF. (4.102): There are two cases. 1. v(tp) = F
Hyp.
2. v?" = -mp
1, (4.101)
3. v{-mp) = T
1, (4.18), (4.13)
4. (-up)" = -i^)
3, (4.101)
p
5. -mp \- -mp v
(4.67)
p
v
6. ip \- {-mp) 1. t)(ip) = T 2.ip
v
2, 4, 5 Hyp.
= ip
1, (4.101)
3. v{-mp) = F
1, (4.18), (4.13)
4. (-.p)" = -n^ip
3, (4.101)
5. 9? hp -n^ip
(4.96)
p
6. 95" \- {~nip)
v
2, 4, 5
(4.103): There are three cases (not mutually exclusive). 1. v(1>) = T
Hyp.
v
2. ip =ip
1, (4.101)
3. t)(p => ip) = T v
4. (v? ^ ip) = ip => ip p
1, (4.17), (4.13) 3, (4.101)
5. i> \- ip ^ i>
(4.67), Th. 142
6. tp" ,ipv \-p (ip =^ ip)v
2, 4, 5, (4.70)
1. v(ip) = F
Hyp.
2. v?" = -. tpy = 4. (
° and stv (i>) h" tpv, but
so, by (4.70), st"(p => ^ ) >-P ¥>" and stv(f => ^ ) I"P ^"- We have 97",^" K (99 => V)™ ^ (4-103), so st B (^ => V) h P (^ =5- VO™ by (4.68). Thus tp => ^ 6 X. To see that X is closed under -i, assume p G X, i.e., st"(ip) h p tp) since st(^) = st(-iy>), and tpv \-p (^tp)v by (4.102), so st w (-.^) h p (-193)* by (4.68), hence -up € X. Since X C FmJ(£), X contains St and is closed under => and ->, we have X = FmJ(£). T h e o r e m 147 (Prepositional Completeness). Assume £ = £(C,T,TZ, rank) is o nice language, n < u, and \-p is provability in FmJ(£) using axioms (HI), (HII), (mil') and rule MP. .For ewn/ ^ € Fm+(£) ! tp is a tautology iff \-p tp. PROOF. By Soundness Th. 141, if h p tp then tp is a tautology. For the converse, assume tp is a tautology, i.e., tpv = tp for every valuation v. Let st(ip) = {ipi,..., ipn}- Then st"(tp) = { ^ 1 , . . . , ip^}. We will prove by induction on * that, if 1 < i < n then ipf ,ipi+i,.. ipn \~p tp for every valuation v : St —> {T, F}. For i = 1 we have tpi,...,^ \-p tp for every valuation w by (4.104). Suppose 1 < % < n + 1 and, as inductive hypothesis, assume that ipi—i,ij>i, ,^« l~p tp p r for every valuation u. We wish to prove \j>%,... ,ipZ, l~ V f° every valuation w if » < n and h p 93 for every valuation v if t = n + 1.
9, COMPLETENESS OP (LI)-(LIII)
20S
Consider an arbitrary valuation v : St —> {T, F} and let w : St —¥ {T, F} be the same as v except that v(ipi-i) =/= w(ipi-i). By the inductive hypothesis we have
By the Deduction Th. 142, plus, in case i < n, the observations that ipf = ipf, ..., tpZ = ipn , we conclude that
(4.105) (4.106)
th — M^tf-i =>
r _ i => V,
or simply h p ^*_i => ^? and h p ^ L i =* ¥> in case t = n + 1. It follows from the choice of w that {ipl-\,ipt-i\ — {ipi-i,~^j>i-i}, and (4.107)
ipi-i =s- p,-.^i-i =4- p P V
by (4.100), so, by (4.105), (4.106), (4.107), and (4.68), we obtain %j}J,...,ijjl\-p(p if i < n and ^ * , . . . , ^ h p ^? if « = n + 1. This completes the proof by induction. Applying the result with i = n +1 yields Hp ip for every valuation v : St —> {T, F}, so h p (p. D 9. Completeness of (LI)-(LIII) In this section we show that Lukasiewicz's axiom schemata (LI)-(LIII) axe complete by deriving (HI), (HII), (Mil') from them. The axiom schemata (HI), (MI), (Mil') are complete, hence (LI)-(LIII) are derivable from (HI), (HII), (Mil'). Theorem 148. Assume C = £(C, IF, TZ, rank) is a nice language, n < u, and \-L is provability in Fm^(C) using axioms (LI), (LII), and (LIU) o-nd rule MP. Let f 6 FmJ(£). ip is a tautology iff\~L (p. (Lukasiewicz [243]) We will show that hL ip whenever p is an instance of (HI), (HII), or (Mil'). Recall that MP is a set of triples. It is a functional relation, in that if {ip, ip, p) and {tp, ip, a) are in M P then p = a. We will treat MP in this proof as a partial function. This means that MP(ip,i/>) is defined to be p if ip = ip p and undefined otherwise. Below we define 36 functions on formulas and draw 33 conclusions listed after the definitions. Our claim, which needs to be confirmed as part of the computations carried out in this proof, is that each of these functions is defined on all inputs and produces the indicated output. For example, our claim concerning fi is that for aH(p,ip,p,a G Fm£(£) the partial (tp p),a) operation MP is defined on the pair of inputs fi(
p) and fi(tp,ip,p), i.e., there is some r 6 Fm^"(£) such that PROOF.
and, furthermore, the output r of M P on those two inputs is f4(p,ip,p,a)
=T= (((ip => p) => {tp => p)) => or) ^ {{ip => ip) => a).
206
4. LOGIC WITH EQUALITY
The range of /i is the set of instances of (LI). Therefore, hL fite,ip,p) for all tp,ip,p 6 Fm+(£). From our claim concerning / 4 we know that / 4 (^J, ,p, , p) te => VO => (W> => P) => if => p)), hiv) hf => p) => (
p),a), fi( (if => {a => p))), f4(a,tp,p,ip =3- (a =3- p))), f6(ip,ip,p,a):=
MP(/ 4 (v?,V,/0,((v=>p)=^)=> (W => P) =^ p,
p,a)), p^ip, {tp^ p) =>a,(ip=>p) =>a,T),f6( f) => p,lfi=> p),f!te,1p,p)),
te, VS P, a)
MP(/i(v? ^(ip^p),ip^(v^-
fiite,
fate,
p), a), f21 (ip, tp, p)),
V>) := MP(MP(/ 23 (v?, "«^, tp, (te => */>) => )),
9, COMPLETENESS OP (LI)-(LIII)
2
2i(
,(ip =^ p) => a,(i/> => p) => cr), fe(, p,cr)),
fso(^)
=> p),f
