**Mathematical Surveys and Monographs**

Volume: 7;
1967;
352 pp;
Softcover

MSC: Primary 20;

**Print ISBN: 978-0-8218-0272-4
Product Code: SURV/7.2**

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**Electronic ISBN: 978-1-4704-1235-7
Product Code: SURV/7.2.E**

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# The Algebraic Theory of Semigroups, Volume II

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*A. H. Clifford; G. B. Preston*

This book, along with Volume I, which appeared previously, presents a
survey of the structure and representation theory of semigroups. Volume II goes
more deeply than was possible in Volume I into the theories of minimal ideals
in a semigroup, inverse semigroups, simple semigroups, congruences on a
semigroup, and the embedding of a semigroup in a group.

Among the more important recent developments of which an extended treatment
is presented are B. M. Sain's theory of the representations of an arbitrary
semigroup by partial one-to-one transformations of a set, L. Redei's theory of
finitely generated commutative semigroups, J. M. Howie's theory of amalgamated
free products of semigroups, and E. J. Tully's theory of representations of a
semigroup by transformations of a set. Also a full account is given of Malcev's
theory of the congruences on a full transformation semigroup.

#### Table of Contents

# Table of Contents

## The Algebraic Theory of Semigroups, Volume II

- TABLE OF CONTENTS v6 free
- PREFACE TO VOLUME II ix10 free
- NOTATION USED IN VOLUME II xi12 free
- CHAPTER 6. MINIMAL IDEALS AND MINIMAL CONDITIONS 118 free
- CHAPTER 7. INVERSE SEMIGROUPS 3956
- 7.1 The natural partial order on an inverse semigroup 4057
- 7.2 Partial right congruences on an inverse semigroup 4259
- 7.3 Representations by one-to-one partial transformations 4764
- 7.4 Homomorphisms of inverse semigroups 5774
- 7.5 Semilattices of inverse semigroups 6481
- 7.6 Homomorphisms which separate idempotents 6582
- 7.7 Homomorphisms onto primitive inverse semigroups 7188

- CHAPTER 8. SIMPLE SEMIGROUPS 8198
- 8.1 Baer-Levi semigroups 8299
- 8.2 Croisot-Teissier semigroups 86103
- 8.3 0-simple semigroups containing 0-minimal one-sided ideals: Gluskin's equivalence 95112
- 8.4 Bisimple inverse semigroups 101118
- 8.5 Any semigroup can be embedded in a simple semigroup. 108125
- 8.6 Any semigroup can be embedded in a bisimple semigroup with identity 112129

- CHAPTER 9. FINITE PRESENTATIONS OF SEMIGROUPS PRODUCTS WITH AMALGAMATIONS AND FREE 115132
- CHAPTER 10. CONGRUENCES 174191
- 10.1 Admissible and normal sets 175192
- 10.2 The principal equivalences of Dubreil. 182199
- 10.3 The reversible equivalences of Dubreil 193210
- 10.4 Principal congruences 198215
- 10.5 Homomorphism theorems for subsemigroups 204221
- 10.6 Weakly permutable relations and the Jordan-Holder Theorem 208225
- 10.7 Congruences on completely 0-simple semigroups 214231
- 10.8 Congruences on a full transformation semigroup 3~x 227244

- CHAPTER 11. REPRESENTATION BY TRANSFORMATIONS OF A SET 249266
- 11.1 Basic definitions 250267
- 11.2 Decomposition of an operand; fully reducible operands and semigroups 256273
- 11.3 Strictly cyclic operands and modular right congruences. 259276
- 11.4 Representations by one-to-one partial transformations. 263280
- 11.5 Irreducible and transitive operands and semigroups. 269286
- 11.6 Various radicals of a semigroup 274291
- 11.7 The normalizer of a right congruence p and the endomorphisms of Sip 279296
- 11.8 Representations by monomial matrices 280297
- 11.9 Other types of representations 284301

- CHAPTER 12. EMBEDDING A SEMIGROUP IN A GROUP 288305
- 12.1 The free group on a semigroup 288305
- 12.2 The general problem of embedding a semigroup in a group 292309
- 12.3 Pták's conditions for embeddability 294311
- 12.4 The construction of quotients 297314
- 12.5 Lambek's polyhedral conditions for embeddability 303320
- 12.6 Malcev's conditions for embeddability 309326
- 12.7 Comparison of Malcev and Lambek systems 319336
- 12.8 Finite sets of equational implications 323340

- BIBLIOGRAPHY 334351
- ERRATA TO VOLUME I 341358
- AUTHOR INDEX 343360
- INDEX 345362
- APPENDIX 351368