**Mathematical Surveys and Monographs**

Volume: 7;
1961;
224 pp;
Softcover

MSC: Primary 20;

Print ISBN: 978-0-8218-0271-7

Product Code: SURV/7.1

List Price: $49.00

AMS Member Price: $39.20

MAA Member Price: $44.10

**Electronic ISBN: 978-1-4704-1234-0
Product Code: SURV/7.1.E**

List Price: $49.00

AMS Member Price: $39.20

MAA Member Price: $44.10

# The Algebraic Theory of Semigroups, Volume I

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

The material in this volume was presented in a second-year graduate course at Tulane University, during the academic year 1958–1959. The book aims at being largely self-contained, but it is assumed that the reader has some familiarity with sets, mappings, groups, and lattices. Only in Chapter 5 will more preliminary knowledge be required, and even there the classical definitions and theorems on the matrix representations of algebras and groups are summarized.

#### Table of Contents

# Table of Contents

## The Algebraic Theory of Semigroups, Volume I

- TABLE OF CONTENTS v6 free
- PREFACE ix10 free
- NOTATION USED IN VOLUME I xiii14 free
- CHAPTER 1. ELEMENTARY CONCEPTS 118 free
- 1.1 Basic definitions 118
- 1.2 Light's associativity test 724
- 1.3 Translations and the regular representation 926
- 1.4 The semigroup of relations on a set 1330
- 1.5 Congruences, factor groupoids and homomorphisms 1633
- 1.6 Cyclic semigroups 1936
- 1.7 Units and maximal subgroups 2138
- 1.8 Bands and semilattices; bands of semigroups 2340
- 1.9 Regular elements and inverses; inverse semigroups 2643
- 1.10 Embedding semigroups in groups 3451
- 1.11 Right groups 3754
- 1.12 Free semigroups and generating relations; the bicyclic semigroup 4057

- CHAPTER 2. IDEALS AND RELATED CONCEPTS 4764
- 2.1 Green's relations 4764
- 2.2 D-structure of the full transformation semigroup J[sub(x)] on a set X 5168
- 2.3 Regular D-classes 5875
- 2.4 The Schützenberger group of an H-class 6380
- 2.5 0-minimal ideals and 0-simple semigroups 6683
- 2.6 Principal factors of a semigroup 7188
- 2.7 Completely 0-simple semigroups 7693

- CHAPTER 3. REPRESENTATION BY MATRICES OVER A GROUP WITH ZERO 86103
- CHAPTER 4. DECOMPOSITIONS AND EXTENSIONS 121138
- 4.1 Croisot's theory of decompositions of a semigroup 121138
- 4.2 Semigroups which are unions of groups 126143
- 4.3 Decomposition of a commutative semigroup into its archimedean components; separative semigroups 130147
- 4.4 Extensions of semigroups 137154
- 4.5 Extensions of a group by a completely 0-simple semigroup; equivalence of extensions 142159

- CHAPTER 5. REPRESENTATION BY MATRICES OVER A FIELD 148165
- APPENDIX A 207224
- BIBLIOGRAPHY 209226
- AUTHOR INDEX 217234
- INDEX 219236