O R D I N A L ALGEBRAS BY
ALFRED TARSKI Professor of Mathematics, University of California, Berkeley
WITH APPENDICES B Y
CHEN CHUNGCHANG Instructor i n Mathematics, Cornell University AND
BJARNI JONSSON Assistant Professor of Mathematics, Brown University
1 9 5 6
NORTH-HOLLAND P U B L I S H I N G COMPANY AMSTERDAM
INTRODUCTION An important task in creating the general theory of binary relations is to develop the arithmetic of relation types, i.e., to study operations by means of which the isomorphism types of complicated relations can be obtained from those of simpler ones. This arithmetic may eventually become a powerful instrument which will give us a better insight into the structural variety of relations. At present, however, it is still in the initial stages. All that could be found until recently in the literature of the subject were: the definitions of fundamental arithmetical operations ; the most elementary and obvious consequences of these definitions; some scattered arithmetical results of a deeper character concerning order types (i.e. types of simply ordering relations), which form a rather narrow class of relation types; and, finally, the detailed development of the arithmetic for a still narrower class of relation types, in fact, for ordinals (i.e., types of well ordering relations). 1) For non-binary relations the arithmetic of relation types practically does not exist. One of the fundamental arithmetical operations on relation types is relational addition. This operation is performable on an arbitrary system of relation types which are indexed by the elements forming the field of a given relation; therefore we speak of the addition of relation types over a given relation. If the field of this relation has just two different elements, relational addition ~
Fundamental arithmetical operations on order types and ordinals 1) were f i s t defined and discussed by G. Cantor. The definitions of these operations, together with various consequences and a systematic development of the arithmetic of ordinals, can be found in various treatises of set theory, e.g. in [6], [13], and [15]. The operations were extended to arbitrary relation types by A. N. Whitehead and B. Russell; cf. [20], vol. 2, pp. 291 ff. Some new notions in this domain were introduced by G. Birkhoff; see [2], pp. 7 ff. (where references to earlier papers can also be found).
2
INTRODUCTION
becomes a binary operation. Actually there are three distinct binary operations which can thus be obtained as particular cases of general relational addition; in fact, cardinal addition, square addition, and ordinal addition. Among these three operations, ordinal addition proves to be the only one which presents a considerable mathematical interest. The main purpose of this monograph is just the development of the theory of ordinal addition for arbitrary relation types. It will be seen from our discussion that a substantial body of results is now available in this domain. Most of the results we shall establish have very simple formulations, but their proofs are usually far from being obvious and sometimes are really involved. Also various problems in the theory of ordinal addition, with equally simple formulations, are still open, and no mechanical method for solving such problems will ever be found. We are primarily interested in binary ordinal addition and its immediate recursive generalization, the ordinal addition of finite sequences. It turns out, however, that the study of finite sums is greatly facilitated by application of certain results concerning infinite sums, in fact, ordinal sums of simple infinite sequences. For this reason we discuss here sums of simple infinite sequences as well, but only insofar as the properties of these sums are involved in the study of finite addition. Another auxiliary operation which plays some part in our discussion is the unary operation of conversion. The method applied in this monograph in the development and presentation of the theory of ordinal addition is the abstract algebraic method which was applied for analogous purposes in an earlier work of the author, [17] (see bibliography at the end of the monograph). Instead of studying relation types and their ordinal sums directly, we concern ourselves in Chapter 1 with abstract algebraic systems formed by a set A of arbitrary elements, an operation 2 on finite and simple infinite sequences of elements of A , an operation + on couples of elements of A , an operation * on single elements of A , and a distinguished element 0 of A.
INTRODUCTION
3
w e single out a certain class of such algebraic systems called the ordinal algebras, by stipulating that the elements and operations which constitute these systems satisfy some relatively simple postulates of an arithmetical character. We then develop the arithmetic of ordinal algebras. Loosely speaking, from postulates defining ordinal algebras we derive consequences concerning elements and fundamental operations of an arbitrary ordinal algebra (regarded as fixed throughout the discussion), without going into the proper algebraic study of ordinal algebras, i.e., without discussing relations between various algebras, methods of constructing these algebras, etc. I n Chapter 2 we show that the postulates defining ordinal algebras are satisfied if we take for A the set of all reflexive relation types (i.e. isomorphism types of reflexive relations), for 2 and + the familiar operations of ordinal addition, for * the operation of conversion, and for 0 the type of the empty relation. I n other words, the set A of reflexive types, with 2, +, *, and 0 thus specified, forms an ordinal algebra (and the same holds for various subsets of A , e.g., for the sets of all types of partially ordering and simply ordering relations). Hence it follows immediately that all the results obtained in Chapter 1 apply to arbitrary reflexive types and their ordinal sums. With certain modifications in the notions involved the results can be extended to arbitrary relation types. Many results in this monograph were originally established by Adolf Lindenbaum and partly by the author for order types and subsequently extended by the author to arbitrary relation types; they were first stated without proof in [S]. Most of the remaining results appear here in print for the fist time; this does not apply, of course, to various familiar elementary theorems of an auxiliary character. The algebraic treatment of the subject is new. This treatment has necessitated providing virtually new proofs for almost all the results discussed. 2, The following theorems and corollaries in this work were established 8) for order types by Lindenbaum around 1926; 1.27-1.29, 1.37, 1.38 and its converse, 1.39, 1.46, and 1.49-1.51. Theorems 1.40-1.44 were obtained in
4
INTROD17CTION
The monograph is provided with two appendices. Appendix A, by Chen-Chung Chang, contains the solution of some arithmetical problems formulated in Chapter 1. Appendix B, by Bjarni Jbnsson, concerns the general operation of relational addition. It introduces the notion of a relation type which is indecomposable relatively t o a given set A of relations (or relation types), i.e., of a type which cannot be represented as a sum of types over a relation belonging to the set A , unless all the terms of the sum except one are equal to 0. The main result of the appendix is a unique decomposition theorem by which, relatively to any given set A of relations satisfying certain general conditions, every reflexive relation type can be uniquely represented as a sum of indecomposable types over a relation belonging to the set A . This result generalizes three special unique decomposition theorems which were previously found (by Chang, Jhnsson, and the author of the monograph) and which can be obtained from the general result by taking for A the set of all cardinal relations or the set of all square relations or, finally, the set of all simply ordering relations. It may be mentioned that the first two of these special theorems are of crucial importance for the theories of cardinal addition and square addition. As a consequence of those theorems (and the commutativity of the operations involved), both cardinal addition and square addition easily reduce to the addition of cardinal numbers; hence the theories of these two operations-as opposed to the theory of ordinal addition - have a very elementary character and are indeed mathematically trivial. The author takes this opportunity to express his warm gratitude to Dr. Chang and Professor Jbnsson for enriching the contents of the same period by the author. Most of these results are stated without proof (and sometimes in a slightly different formulation) in [ 8 ] , pp. 320 f. The proofs of 1.27, 1.37, and 1.46 can be found in [14], pp. 3-7, and [15], pp. 167-169; the proofs of other theorems in this group were not previously published. For those of the remaining results which are not entirely due to the author the question of authorship will be cleared up in remarks or footnotes accompanying the theorems involved.
INTRODUCTION
5
the monograph with their interesting results. Warm thanks are also due to Dr. Chang and Mr. William Hanf for their help in preparing this book for publication. The technical work on the book was largely done during the period when the author, Hanf, and, for a short time, Chang were engaged in a research project in the foundations of mathematics sponsored by the National Science Foundation, U.S.A.
CHAPTER 1 ORDINAL ALGEBRAS AND THEIR ARITHMETIC In the following discussion we shall use some fundamental notions of general set theory; thus, e.g., the notions of an element and a subset (the membership relation E and the inclusion relation C),and those of union and intersection. The notion of an ordinul, the relations < and < between ordinals, and the operations + and . on ordinals are assumed to be known. The ordinals will be represented by variables x , 1,p, v, _...In this chapter we shall deal exclusively with finite ordinals (which may be identified with natural numbers) and with the smallest infinite ordinal, w . In formulating theorems involving some ordinals p, v, ..., the assumptions p ~ wv , ~ w... , will usually be omitted; the formulas p < o , v < w , ... will be used to express the fact that the ordinals p, v, ... are finite. In the sums p + v and products p - v involved, p will always be a finite ordinal ; as is well known, p + w = w for every p < w , and p . w = w for every p < o provided p # 0. In a difference p - v the ordinals p and v are assumed to satisfy the formula v
1. By hypothesis we have
(
a.p+[[b.(V-p-l)J.p=b.Y+b.[(~-p-l)~p]= (15) =b*[v+(v-p- l)-p]= [b-(p+l)]*(v-p).
Moreover, the hypothesis implies by 1.49
a+b=b+a;
hence by 1.13(i) (16)
a + b * ( v - p - l)=b.(v-p-
l)+a.
From (15) and (16) we derive by 1.13(ii) (17)
[u+b*(v-p- 1)]-p= [ b * ( p +l)].(v-p).
It is well known that, whenever p and v are relatively prime, p and v-p are also relatively prime. I n view of (12) we have p
+ (v-p)
1, we have
p+fp++) E S -
(2,
Hence we conclude that the operation + is always performable in the domain of relation types; i.e., for any two types a and P, a + p is a uniquely determined relation type. I n an entirely analogous way we define the ordinal sum Ex , and (0,2); put and L Y ~ + ~ = Ofor x < w .
a,,=y=z(R), al=B=t(S),
As is easily seen, the types a,,, hypothesis of 1.1(111), i.e., &
[F(uk)xF(uZ)l)*
Using B.l(iii), B.2(i), and 2.1(i), we see that, if (k, 1) E S and k # l , then Ti
[(F( uk) x
( uZ)l = =
=
CF
IF
( uk)
x
( ul)l
CFVi)n F ( Uk)l x [ F (Ti)n F ( UJl F ( T i n U , ) x F ( T , n UZ),
and we conclude by (1) that
Ti = s , s (Tin uJ* Next we prove a refinement theorem which will subsequently be used to obtain unique decomposition theorems for the addition of reflexive relations (B.6) and the addition of reflexive relation types (B.9).
THEOREM B.5 [GENERAL REFINEMENT THEOREM FOR RELATIONAL ADDITION]. Let K be a set of reflexive relations with the following properties (a) there exists a n R E K such that R#O; (b) if R E K and S i s isomorphic to a subrelation of R , then S E K ; (c) if X E K for every finite subrelation S of R, then R E K ; (d) if R E K , Si E K for every i E F ( R ) , and S,)(Xj whenever a', j E F ( R ) and i#j, then c.,Si E K.
102
APPENDIX B
Suppose R is a reflexive relation, I is a non-empty set, and Si E K for every i E I . If R = Zkm8iQi,k for every i E I , and if Qi,k)( Qi,l whenever i E I , k, 1 E F ( S J and k # l , then there exist reflexive relations T , Wi,kwith i E I and k E F(S,), and U , with f E F ( T ) which satisfy the following conditions : (i)
T E K , and WiVkE K for all i E I a.nd k E F(S,); (ii) R=C,.,U,, and U,#O for every f E F ( T ) ; (iii) T = for every i E I ; (iv) QiSk= zj,w,,U, for all i E I and k E F(S,); (v) U j ) ( U,, whenever f , g E F ( T ) and f # g ; (vi) WiSk)(Wi,, whenever i E I , k, E E F(S,), and k f l . PROOF:Observe that by B.3 all the relations QiSkwith i E I and k E F(S,) are reflexive. Consider the set C of all functions f on I such that f ( i ) E F(S,) for every i E I, and for each f E C let
z.S6Wd.k
U , = n i e r Qi.m-
(1)
Let the relation V and the set D be defined by the conditions: (2)
( f , g) V if { every i I. E
and only if f , g E C and ( f ( i ) , g ( i ) ) € 8 , for
E
f
(3)
E
D if and only if f E C and U,#O.
Also let
T = V n (Dx D),
(4)
and for each i E I and k by the condition: (5)
(f4g)
E
E
F ( S J let Wi.k be the relation defined
WiaEif and only if ( f , g ) E T and f ( i ) = g ( i ) = k .
By our hypothesis we have = ' k e P ( S 4 l 'k.i&
for every i (6)
E
U(k.I)eSi.k+Z
[F(&6.k)
x F(Qi,t)l
I and, consequently,
= nieI
(ukeP(S')
Qik
U(k.Z)eSr,k+2 [F(&i.k)
F(&i.t)l)
DECOMPOSITION THEOREM FOR RELATIONAL ADDITION
For each i
E
103
I and (k, Z) E S let ~ Yi.k.l
=
x
LP(&i.k)
(Qi.t)1
9
and use B.2(i),(ii) to infer that Y ; , k , k = Qi,k
Y4,k.I =
F (QiSk) X F
for every i E I and k E E”(s,), whenever i €1,0,I> E Si, and k # 1.
(Qi,$)
By (2) and (6) we therefore have
R = U(r.o)sv nitr Yi.tci),m or, equivalently,
R = Ujtc nisIQijci)u U E T , P(U,)x F(U,) R, P ( K ( , , )x F(Jc(,,) c R, ( h ( f ) h(gD , E
c
are equivalent, so that h maps T isomorphically onto Y . The proof has thus been completed. Passing now to relation types we f i s t state, in B.7 and B.8, several elementary properties of relational addition of types.
THEOREM B.T. (i) ~ R O = z i , O c r i = O . If j E F(R), and if a,=O whenever i (ii)
xaa(
E
F(R) and i# j , then
= ai.
(iii) If j E F(R), then cri is a subtype of z , R a i . then is a (iv) I f , for every i E F(R)lL Y ~ is a subtype of #Ii, subtype O f z j , R b i ’ ( v ) If T=z:,.,Si, and if Si)(Si whenever i , j E F(R) and i # j , then G.R Z,s$~j = 29.T L*i* (vi) If B i s the set of all elements i E F(R) such that ai#O, and if S = R n ( B x B ) , then (vii)
z,Bmi
x,R~6=x,scri.
I X * ( Z < , R B i )= z i , B
(@. B 6 ) ’
PROOF:Parts (i), (ii), (v), (vi), and (vii) axe easy consequences of B.1, while (iii) and (iv) readily follow- from the definitions of the notions involved.
THEOREM B.8. (i) 0, 1 E RT. E RT if and only if cri E RT for every i E F(R). (ii) If z(R) E RT, then zi,R1=z(R). (iii) (ic) a . O = O . c r = O and rn.l=cr. (v) If cr E RT, then l.cr=a. PROOF:obvious,
x,Rai
We now obtain the main result of this appendix:
THEOREM B.9
[UNIQUE DECOMPOSITION
THEOREM FOR RELA-
110
APPENDIX B
Let A be any set of reflexive types satisfying the TIONAX, ADDITION]. following conditions : (a) 1 E A ; (b) if a E A and p i s a subtype of a , then /3 E A ; E A for every finite subtype ,L? of a , then a E A ; (c) if (d) if t ( R )E A and a{ E A for every i E P ( R ) ,then EA. Under these assumptions every reflexive type a can be represented in the form a = Xi.RBi
x,R~i
where t ( R )E A , and b5 E I ( A ) for every i E F ( R ) . This representation is unique in the following sense: if is another representation of the S a m kind, then there exists a function h which maps R ismorpitically onto S in such a way that pi= yh,{)fm every i E F(R). PROOF: by B.6. Using the terminology introduced in Chapter 3 , p. 82, we can formulate the last part of the preceding theorem more simply: if 01
=2t.R
Pi = 25.8 ~j
are two representations of LS: of the kind discussed, then the systems
