Automata, Languages and Machines. Volume B. (Pure & Applied Mathematics)

AUTOMATA, LANGUAGES, AND MACHINES VOLUME B

Pure and Applied Mathematics A Series of Monographs and Textbooks Editors Samuel Eilrnberg and Hyman Earr Columbia University, N e w York

RECENT TITLES

T. BENNYRUSHING.Topological Embeddings JAMES W. VICK.Homology Theory : An Introduction to Algebraic Topology E. R. KOLCHIN. Differential Algebra and Algebraic Groups J. JANUSZ. Algebraic Number Fields GERALD Introduction to the Theory of Entire Functions A. S. B. HOLLAND. WAYNE ROBERTS A N D DALEVARBERG. Convex Functions A. M. OSTROWSKI. Solution of Equations in Euclidean and Banach Spaces, Third Edition of Solution of Equations and Systems of Equations H. M. EDWARDS. Riemann’s Zeta Function SAMUFLEILENBERG. Automata, Languages, and Machines: Volumes A and B M o m s HIRSCH A N D STEPHEN SMALE. Differential Equations, Dynamical Systems, and Linear Algebra WILHELM MAGNUS.Noneuclidean Tesselations and Their Groups FRANCOIS TREVES. Basic Linear Partial Differential Equations WILLIAM M. BOOTRBY. An Introduction to Differentiable Manifolds and Riemannian Geometry BRAYTON GRAY.Homotopy Theory : An Introduction to Algebraic Topology ROBERT A. ADAMS.Sobolev Spaces JOHN J. BENEDETTO. Spectral Synthesis D. V. WIDDER. The Heat Equation IRVING EZRA SEGAL. Mathematical Cosmology and Extragalactic Astronomy J. DIEUDONN~. Treatise on Analysis : Volume 11, enlarged and corrected printing ; Volume I V WERNER GREUB,STEPHEN HALPERIN, AND RAYVANSTONE. Connections, Curvature, and Cohomology : Volume 111, Cohoniology of Principal Bundles and Homogeneous Spaces

In fireparation I. MARTINISAACS. Character Theory of Finite Groups K. D. STROYAN A N D W. A. J. LUXEMBURG. Introduction to the Theory of Infinitesimals JAMES R. BROWN. Ergodic Theory and Topological Dynamics CLIFFORD A. TRUESDELL. A First Course in Rational Continuum Mechanics : Volume 1, General Concepts MELVYN BERCER. Nonlinearity and Functional Analysis : Lectures on Nonlinear Problems in Mathematical Analysis

AUTOMATA, LANGUAGES, AND MACHINES VOLUME B

Samuel Eilenberg COLUMBIA UNIVERSITY NEW YORK

With two chapters by Bret Tilson CITY UNIVERSITY OF NEW YORK QUEENS COLLEGE NEW YORK

ACADEMIC PRESS

New York San Francisco London

A Subsidiary of Harcourt Brace Jovanovich, Publishers

1976

COPYRIGHT 0 1976, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY B E REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York. New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NWl

Library of Congress Cataloging in Publication Data Eilenberg, Samuel. Automata, languages, and machines. (Pure and applied mathematics: a series of monographs and textbooks) Vol. B includes two chapters by Bret Tilson. Includes bibliographies. 2. Formal 1. Sequential machine theory. 1. Title. 11. Series. languages. 3. Automata. QA3P8 vol. 59 51 0'3s 1629.8'91 ] 72-88333 ISBN 0-12-234002-7 (pt. B.)

PRINTED IN THE UNITED STATES OF AMERICA

Contents PREFACE . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

CHAPTER I Transformation Semigroups 1. Semigroups. Monoids. and Groups 2

.

3.

4.

.

5 6. 7. 8 9 10.

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Transformation Semigroups . . . . . Examples of Transformation Semigroups Coverings . . . . . . . . . . . . . Coverings of Semigroups . . . . . . Inclusions and Restrictions . . . . . Isomorphisms and Equivalences . . . Join. Sum. and Direct Product . . . . Some Simple Inequalities . . . . . . The Wreath Product . . . . . . . . References . . . . . . . . . . . . .

1 3

. . . . . . . . . . . . . . .

5 8 12 14 16 18 22 26

. . . . . . . . . . . . . . .

32

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER I1 Decomposition Theorems 1. Decompositions . . . . . . . . . . . . . . . 2 . Decomposition of Groups . . . . . . . . . . 3 Some UsefuI Decompositions . . . . . . . . . 4 T h e Krohn-Rhodes Decomposition . . . . . 5 Comments on the Proof . . . . . . . . . . . 6. Height. Pavings. and Holonomy . . . . . . 7. The Holonomy Decomposition Theorem . . 8 . Proof of Proposition 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Examples References . . . . . . . . . . . . . . . . .

. . .

V

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 34 36 39 43 43 46 48 51 57

Contents

vi

CHAPTER I11 Transformation Semigroups (continued) 1. 2. 3. 4. 5.

Classes and Closed Classes . . . . . . . . . . . . . . . . . . . . . Sinksinats . . . . . . . . . . . . . . . . . . . . . . . . . . . Transitivity Classes . . . . . . . . . . . . . . . . . . . . . . . . Idempotents in Semigroups . . . . . . . . . . . . . . . . . . . . Idempotents in a ts . . . . . . . . . . . . . . . . . . . . . . . . 6 . Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Closed Classes Containing 2' . . . . . . . . . . . . . . . . . . . . 8 The Derived ts and the Trace of a Covering . . . . . . . . . . . . . 9. The Delay Covering . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

59 62 66 67 69 72 74 76 80

86

CHAPTER IV Primes 1. The Exclusion Operator

2. 3. 4. 5. 6. 7. 8.

...................... .. .. .. .. . . .. . . . . ..

Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . The Low Primes . . . . . . . . . . . . . . . . . . . . . . . T h e Primes C and C' . . . . . . . . . . . . . . . . . . . . . The Primes F, 2, F'. and z' . . . . . . . . . . . . . . . . . . . Switching Rules . . . . . . . . . . . . . . . . . . . . . . . Summary and Open Problems . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

87 88 90 94 96 100 102 105 108

CHAPTER V Semigroups and Varieties 1

.

2. 3. 4. 5. 6. 7. 8

.

9. 10. 11. 12.

Varieties of Semigroups and Monoids . . . . . . . . . . . . . . . . Varieties Defined by Equations . . . . . . . . . . . . . . . . . . . Examples of Ultimately Equational Varieties . . . . . . . . . . . . . Semidirect Products . . . . . . . . . . . . . . . . . . . . . . . . Varieties V W . . . . . . . . . . . . . . . . . . . . . . . . . . Varieties vs. Weakly Closed Classes . . . . . . . . . . . . . . . . . Closed Varieties . . . . . . . . . . . . . . . . . . . . . . . . . Examples of Closed Varieties . . . . . . . . . . . . . . . . . . . . Triple Products . . . . . . . . . . . . . . . . . . . . . . . . . . G-Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Tabulation . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 112 116 123 129 132 135 138 142 144 150 152 156

Contents

vii

CHAPTER VI Decomposition of Sequential Functions 1. 2. 3 4. 5. 6.

.

7.

Syntactic Invariants of Sequential Functions . . . . . . . . . . . Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . Parallel Composition . . . . . . . . . . . . . . . . . . . . . . . Examples of Decompositions . . . . . . . . . . . . . . . . . . . . T h e Function ......................... Varieties of Sequential Functions . . . . . . . . . . . . . . . .

. .

. .

157 162 163 168 174 178 181

........... . . . . . . . . . . . . ........... . . . . . . . . . . . . ........... . . . . . . . . . . . .

185 188 192 197 199 202

s

CHAPTER VII Varieties of Sets

. . . .

1 2. 3 4 5. 6

Syntactic Semigroups . . . . . . . . . . . Syntactic Semigroups and Recognizable Sets Varieties of Sets . . . . . . . . . . . . . Proof of Theorems 3.2 and 3.2s . . . . . . Operations on Varieties . . . . . . . . . . The Syntactic tm and ts of a Set . . . . .

. . . . . .

CHAPTER VIII Examples of Varieties of Sets

. . . . . .

1 2 3 4 5 6. 7

8.

9. 10

.

General Comments . . . . . . . . . . . . . . . . . . . . . . . . Finite and Cofinite Sets . . . . . . . . . . . . . . . . . . . . . . Finitely Generated Varieties . . . . . . . . . . . . . . . . . . . . T h e Variety D . . . . . . . . . . . . . . . . . . . . . . . . . . The Variety b . . . . . . . . . . . . . . . . . . . . . . . . . . Locally Testable Sets . . . . . . . . . . . . . . . . . . . . . . . . A Theorem on Graphs . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 6.5 . . . . . . . . . . . . . . . . . . . . . . . T h e *-Variety ......................... $-Groups ............................ References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207 208 210 214 216 218 222 228 232 238 245

CHAPTER IX Aperiodicity

. . .

1 2 3

Recognizable Sets and Sequential Functions The Concatenation Product Schiitzenberger’s Theorem . . . . . . . . .

. . . . . . . . . . . . .

....................

.............

247 249 253

viii

. . 6. 7.

Contents

..................... ....................... The Variety B. . . . . . . . . . . . . . . . . . . . . . . . . . . The Variety Al . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

The Brzozowski Hierarchy

5

Sn. Are +-Varieties

256 259 261 263 268

CHAPTER X Unitary-Prefix Decompositions

. . .

1 2 3 4. 5. 6

.

Unitary-Prefix Decompositions A Decomposition . . . . . . Two Examples . . . . . . Iterated Decomposition . . . Periods of Monoids . . . . . Proof of Theorem 5.2 . . . . References . . . . . . . .

. . . . . . ....... . . . . . . . ....... ....... ....... . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . . . . . . .......... . . . . . . . . . . .......... .......... .......... . . . . . . . . . .

269 272 274 277 279 282 285

CHAPTER XI Depth Decomposition Theorem by Bret Tilson

. . . .

1 2 3. 4 5 6.

Basic Orderings in Semigroups . . . . . . . . . . . . . . . . . . The Depth Decomposition Theorem . . . . . . . . . . . . . . . The Rees Matrix Semigroup . . . . . . . . . . . . . . . . . . . . The Reduction Theorem . . . . . . . . . . . . . . . . . . . . . Proof of Proposition 2.2 . . . . . . . . . . . . . . . . . . . . . . Comparison with Holonomy Decomposition . . . . . . . . . . . . References ...........................

. .

.

287 295 297 300 304 308 311

CHAPTER XI1 Complexity of Semigroups and Morphisms by Bret Tilson

1. 2 3 4 5. 6 7 8 9

. . . . . . .

Definition and Basic Properties . . . . . . . . . . . . . . . . . The Standard Complexity . . . . . . . . . . . . . . . . . . . . . Complexity of Morphisms . . . . . . . . . . . . . . . . . . . . . Morphism Classes Defined by S-Varieties . . . . . . . . . . . . T h e Main Theorems of Complexity . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complexity of Projections . . . . . . . . . . . . . . . . . . . . . The Derived Semigroup of a Morphism . . . . . . . . . . . . . The Rhodes Expansion . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

313 320 326 331 337 339 351 356 361

ix

Contents 10. 11 . 12. 13. 14

.

Proof of the Ideal Theorem . . . . . . . . . . . Construction of the Rhodes Expansion . . . . . SIsFine . . . . . . . . . . . . . . . . . . . Proof of Property (9.6) . . . . . . . . . . . . . Problems, Conjectures. and Further Results . . . References . . . . . . . . . . . . . . . . . . .

INDEX

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

362 367 372 375 379 382

385

This Page Intentionally Left Blank

Preface

The objective of this volume is to study, by algebraic methods, the properties of recognizable sets (i.e. sets recognized by finite state automata) and of sequential functions. The algebra is introduced by means of the following device. Let A be a recognizable subset of Z* where Z is a finite alphabet, and let d =( Q A , i, T ) be the minimal automaton of A. Each letter B E C defines a partial function (the automaton need not be complete!) Q A Q A . These partial functions generate a subsemigroup SA of the finite monoid of all partial functions QA -+ QB. This semigroup SA is called the syntactic semigroup of A and the pair TSA = (QA, SA)is called the syntactic transformation sem2roup of A. If we adjoin the identity transformation of QA4to SA, we obtain the syntactic monoid M A and the syntactic transformation monoid TMA = ( Q A , MA). If we start with a sequential function f : C" -+ P, we apply the same procedure to the minimal sequential machine A? = (Qr , i, A): 2 - 'I off. This yields syntactic invariants off. Clearly, if interesting information about A and f is to be gleaned out of the syntactic invariants, we must know a good deal about these algebraic objects. This puts the spotlight on transformation semigroups and transformation monoids and also on semigroups and monoids, with everything in sight assumed to be finite. As expected, a good deal of more or less new algebra will have to be used, and this algebra is developed in Chapters I-V. Chapter I introduces ts's (i.e. transformation semigroups) and tm's (i.e. transformation monoids) and defines basic concepts for dealing with them. Among these are -+

X 0, then C ( p , r ) is a ts (but not a tm) and = Z ( p , r )is the cyclic monoid with stem r and period p . The action semigroup S ( p , rof ) C ( p , ris) the cyclic semigroup with generator a and relation ar = a r + p . With the exception of the is the ts defined by S t p , r fIn . the exceptional case (p, Y ) = (1, l), C(p,r) case (9, r ) = (1, l ) , C(l,l)is the ts C whose action semigroup is the semigroup 1 containing a single element. Thus C,,,,, and C,,,,, have isomorphic semigroups. The family A for C ( p , rconsists ) of the empty set, all singletons, and sets

ai= { a j I i s j < r + p }

for 0 5 i 5 Y . Note that ao is interpreted as the state 1. Each set ai for 0 5 i < r is paved by the two sets {d, ai+l}and the holonomy ts is Had= 2 with no transformations. The set a, = {a+,. . . , ar+p--l}is paved by its singletons and Ha, = 2,. Thus the Holonomy Decomposition Theorem yields C(p,r)

< zp zr 0

This decomposition is far from being the best possible one. We shall now show by direct arguments how a “better” decomposition can be achieved. First we note that there exists an injective morphism S(p,r)

+

ZpX

S(1.r)

9. Examples

53

of monoids. The morphism maps the generator 0 into (t,7) where T is the generator of Z p and 7 is the generator of S(l,r). This implies C(p.r)


0, then Proposition 3.3 can be applied with the state 1 playing the role of the state p in the proposition. Removing the state 1 we obtain ,, the states renamed. Thus Proposition 3.3 yields the the ts C ( , ~ , - with inequality ~ ( l , r ) e(l.r-1) 0 C


1. Then

I

PROPOSITION 4.4. Let p: T -+S be a surjectiwe morphism of semigroups. For each idempotent e in S there is an idempotent e' in T such that e = e'p

Since p is surjective, ep-' is a non-empty subsemigroup of T. Thus by Proposition 4.1, ep-' contains an idempotent I Proof.

PROPOSITION 4.5. Let p: T --+ S be a surjective morphism of semigroups. For each monoid (or group) S' in S there exists a monoid (OY group) T in T such that S' = T'p. Proof. Let T' be a subsemigroup of T of lowest possible cardinality such that T'p = S'. Such a subsemigroup exists since S'p-lp = S' and

5. ldempotents in a

69

ts

S'v-' is a subsemigroup of T. Let e be the unit element of S'.By Proposition 4.4, there exists an idempotent f in T' such that fv = e. Since

( f T Y ) v= (fv)(T'v)(fv) = eS'e = S' and since f T ' f c T' it follows from the minimality of T' that f T Y = T'. Thus T' is a monoid with f as unit element. Now assume that S' is a group. Let t E T'. Then

( T ' t )v = ( T ' v ) ( t T )= S ' ( t v ) = S' and since T't c T' it follows from the minimality of T' that T't Similarly, tT' = T'. Thus T' is a group, by Proposition 1,l.l I

=

T'.

COROLLARY 4.6. If S < T where T is a semigroup and S is a group (or a monoid), then there exists a group (or a monoid) T' in T and a surjective morphism v: T' -+ S I


sn)

* * *

Y , is an injective function and therefore X n < Yn

I

We note that X , is only vaguely related to X as in the definition of

X, , Q and S are used only as sets. The semigroup structure of S and the action of S on Q will however be used to define the nth delay covering

XdXn *tI

8,:

Qn+

Q l w l I n - 1

w8,=QE

if

lwl=n

The fact that the relation 8, is surjective is clear. T h e inclusion 6,s c $6,

9. The Delay Covering

83

is easily verified and shows that 6, is a relational covering. Observe that if S = 0 then 52 = 0, X , = Q, and 6, is the identity. There is a natural parametrization associated with this covering, namely, (52,a, /?) with Q and a as used above and with s/? = s^ for all s E 52. I t is with respect to this parametrization that we consider the trace of the covering 6,. First we consider the elements of X , of the form (q, w ) , q E Q, w E a*, 1 w I < n. There is no transformation t E S , such that (q, w ) t c (q, w ) . It follows that

Next we consider w = ( s l , . . . , s,) E Q,. Let t E S, be a transformation such that wt = w. Write t as a composition t = tAl . . . f, with t , , . . . , t , E S. From the definition of X , it follows that wt is the terminal segment of length n of the word (sl, . . . , s,, t , , . . . , t,) in Q+. Thus wt = w holds iff

Consequently if we define Rw =

{X

Ix

E

52+, w x E Q+w}

we find that the trace

Tr,

=

(!a,SW)

is represented by (Qg,R,). PROPOSITION 9.4.

If n 2 card S,

there exists an idempotent e E

then for each w E Qf,

I w I = n,

S such that

Proof. Since I w I 2 card S , Proposition 9.1 applies. From all the factorizations w = uv that satisfy the assertion of Proposition 9.1, choose the one with I v I minimal, and let e be the idempotent such that iie = Q.

Let x E R,. We show that v is a terminal segment of x so that xv-l is a word in 52". Indeed, since wx E Q+w, we have wx = yw = yuv for some y E Q+, so either v is a terminal segment of x or vice versa.

84

Assume 1 v I > I x

Ill. Transformation Semigroups (continued)

I. Then

v = zx for some z E

9+and

w x = yuzx

Cancellation of x yields w = yuz with j5ie = jjii and 1 z I < I v I, a contradiction to the minimality of I w I. Therefore v must be a terminal segment of x. Now w x = yuv implies wxv-1 = yu

(9.2) Since rie = zi, it follows that

with

v: "=

Qe-QZ

{?

if q E Q i i otherwise

Note that Qii = Qiie c Qe and that Qiiv = QE= QZ, so that v is well defined and surjective. We assert that each transfwmation x E R, of Tr, is covered by xq E eSe. Indeed if q6 E Qe then by (9.4) qiifpn = q w x = px-

= qzi(q)B

Since further by (9.3) and (9.2) qzi(xr]) = q i i a Z - l e = pTiZ5-1 = qJii E Qii

9. The Delay Covering

85

it follows that

qzqn = qC(xq)@ = qC(xq)q Thus q f c ( q ) qas required

I

Proposition 9.4 and formula (9.1) combine to give THEOREM 9.5. (Tilson's Trace-Delay Theorem). Let X = (Q, S ) be a ts with S # 0 and let X d8* Y be the nth delay covering of X with n 2 card S. Then for each q E Q y there exists an idempotent e E S such that Tr,<Xe I COROLLARY 9.6. Let X be a weakly closed class in TS containing 1, let X E LX and let X Qa, Y be the n-delay covering of X with n 2 card S, . Then Traced, c X I

As an application consider the case X = [l']. T h e conclusion above asserts that card(@,) 1 for all q E Q,. Thus d, is a covering rather than a relational covering. Thus X < X , and since, by Proposition 9.3, X, E it follows that X E We thus obtain


0. The class (E') consists of all ts's X such that ese c e for all s E Sx and all idempotents e in S,. The inclusion [2, 21 c ( E ' ) holds because

106

IV. Primes

2 , 2 E ( E m ) .T h e opposite inclusion fails even if we restrict ourselves to complete ts's (Exercise 7.4). Most likely the class ( E ) is not finitely generated and also not completely generated ; however, these are open questions. In addition to the 14 primes considered above there are eight intermediate primes (among the divisors of 2'). Each such prime contains P and is contained in P' where P is one of the main primes. There is one prime between 1 and 1' and three between 2 and 2'. Their exclusion classes are considered in Exercise 4.3. In addition there are three primes between E and E and one prime between C and C'. We have no information concerning their exclusion classes ; they are likely to be very complicated. In Section 6 we have shown that the primes

P = F,

Z, F', T

are paired off with the primes

P = 2, c, 2',

C'

in the following sense: a ts X is in (P) iff X I R E ( p ) for each transitivity class R in X . In addition to the primes dividing 2' there are the primes 2, where p is a prime integer. The exclusion classes (2,) are (2,)

For p

=

[T, G

simple of order relatively prime to p ]

= 2,

(2,) = [Z., 2, for p prime, p > 21 T h e situation is somewhat reminiscent of the classification of simple compact Lie groups where there are four main series and five exceptional groups. EXERCISE 8.1. For n 2 1 consider the ts

and let En denote the same ts without the curved arrow. Show that if X where X is En,En*, Fn, or Fn*, then there exists a ts X' such that

X ' C Y,

X'MX

(c, v‘, e ) ] y = (c, ~c = =

=

(c,

(c,

144

V. Semigroups and Varieties

so that y is a morphism. Since (c,

0, e )

= (c, = (c,

d, e)(c, v, e)(c, d, e ) dc evc ed, e )

+

+

it follows that y is injective. Thus y maps H isomorphically onto a group inV I COROLLARY 9.2. Let G be a group in V # S. There exists then an invariant subgroup H of G such that H is isomorphic with a group in V , while G / H is isomorphic with a group in S I PROPOSITION 9.3. Let G be a group such that G exists then an invariant subgroup H of G such that

H

(C) n S = Gs ( F ) n S = RS

(2) n S = (2) n S = Rs r G For the last formula we use the equality (R *: G ) s = Rs rc G implied by Proposition 5.4. All these S-varieties with the exception of G s are closed. For the first four this is clear, while for the last two this follows from Proposition 7.1 since R and R # G are closed M-varieties containing U,.

154

V. Semigroups and Varieties

Next come the S-varieties (P) n S, where P is one of the seven dotted primes. The table is ( 0 ) n s =: 0

(1') n S

=

{@}

(2') n S = D (12.5)

( E ) n s = Z,

(c')n S = ( U , ) n S = G r D (F') nS = R

D

(F) n S = ( U , ) n S = R

#

G

#

D

All these S-varieties are closed. For the first two this is clear. The third and the fourth we treated in Example 8.6. For the last three this follows from Proposition 7.2 since (P) is a closed class containing C. The first four formulae are restatements of earlier results. T h e last three require proofs. We are thus left with the task of proving the last formula of (12.3) and the last three formulae in (12.5). Since the proofs are quite similar, we shall only prove the most involved one, namely the last formula in (12.5). From IV,8, we know

(2')

=

[C']

0

[GI

0

[Z]

Thus Proposition 6.5 applied twice yields

(2.) n S c ([C'] n M)

#

([GI n M)

#

([Z] n S)

Using the earlier results in the tables above this implies

(2.) n S c R # G r D T o prove the opposite inclusion we observe that (2) n S is a closed S-variety. Thus it suffices to prove that (2.) contains R, G, and D. This, however, follows from the fact that (2) contains (F), (C), and

(2.)We note that the tables above contain the varieties (S) n S for the five prime semigroups S and the varieties ( S ) n M for the three prime monoids S.

12. A Tabulation

155

T o the formulae above we add three more formulae dealing with localization G+D=LG

(12.6)

R+D=LR R+G+D=L(R+G)

Indeed, settingV = G, R, or R x G, the tables show thatV = (P) n M where P is C or F o r 2 depending on what V is. Since 2 E (P) and since (P.)= L(P), Proposition 6.3 implies

(P)n S = L(P) n S = L((P) n M ) = LV Equalities (12.6) thus follow by looking up formulae (12.5). It should be observed that if V is any M-variety, then V + (1) and thus by Proposition 5.6

V r L ( 1 ) c L(V+ (1)) Since L ( 1 )

=

b

(12.7)

= V,

= LV

and D c b we obtain V + D cVWZ) c L V

Equations (12.6) thus supply three cases in which the inclusions in (12.7) are equalities. I n VIII,8 we shall prove that (12.8)

JI

+

D = LJ,

No doubt there are other cases in which equality in (12.7) holds. It is an open and interesting question to find criteria for equality in (12.7). It should be noted that ultimately equational descriptions of most of the varieties discussed above are contained in Section 3. EXERCISE 12.1.

Let P be a simple group. Show that

(P) n M =

v

where

V

=