The book "The Algebraic Theory of Semigroups" by A. H. Clifford and G. B. Preston is a comprehensive treatise on the algebraic theory of semigroups. It covers fundamental concepts, structures, and representations of semigroups, including their decompositions, extensions, and representations by matrices over groups and fields. The book is divided into several chapters, each focusing on specific aspects of semigroup theory: 1. ** Elementary Concepts**: Introduces basic definitions, associativity tests, translations, regular representations, semigroups of relations, congruences, factor groupoids, homomorphisms, cyclic semigroups, units, maximal subgroups, bands, semilattices, regular elements, inverses, and embedding semigroups in groups. 2. ** Ideals and Related Concepts**: Discusses Green's relations, the structure of the full transformation semigroup, regular D-classes, 0-minimal ideals, 0-simple semigroups, principal factors, and completely 0-simple semigroups. 3. ** Representation by Matrices Over a Group with Zero**: Explores matrix semigroups over a group with zero, the Rees Theorem, Brandt groupoids, homomorphisms of regular Rees matrix semigroups, and Schützenberger representations. 4. ** Decompositions and Extensions**: Treats Croisot's theory of decompositions, semigroups as unions of groups, decompositions of commutative semigroups, and extensions of semigroups. 5. ** Representation by Matrices Over a Field**: Focuses on representations of semisimple algebras, semigroup algebras, principal irreducible representations, and characters of commutative semigroups. The book aims to provide a systematic exposition of the algebraic theory of semigroups, building on the foundational work of A. K. Suschkewitsch and D. Rees. It is intended for readers with a background in set theory, mappings, groups, and lattices, and includes exercises to reinforce the concepts discussed. The authors acknowledge the contributions of various scholars and provide a detailed bibliography for further reading.The book "The Algebraic Theory of Semigroups" by A. H. Clifford and G. B. Preston is a comprehensive treatise on the algebraic theory of semigroups. It covers fundamental concepts, structures, and representations of semigroups, including their decompositions, extensions, and representations by matrices over groups and fields. The book is divided into several chapters, each focusing on specific aspects of semigroup theory: 1. ** Elementary Concepts**: Introduces basic definitions, associativity tests, translations, regular representations, semigroups of relations, congruences, factor groupoids, homomorphisms, cyclic semigroups, units, maximal subgroups, bands, semilattices, regular elements, inverses, and embedding semigroups in groups. 2. ** Ideals and Related Concepts**: Discusses Green's relations, the structure of the full transformation semigroup, regular D-classes, 0-minimal ideals, 0-simple semigroups, principal factors, and completely 0-simple semigroups. 3. ** Representation by Matrices Over a Group with Zero**: Explores matrix semigroups over a group with zero, the Rees Theorem, Brandt groupoids, homomorphisms of regular Rees matrix semigroups, and Schützenberger representations. 4. ** Decompositions and Extensions**: Treats Croisot's theory of decompositions, semigroups as unions of groups, decompositions of commutative semigroups, and extensions of semigroups. 5. ** Representation by Matrices Over a Field**: Focuses on representations of semisimple algebras, semigroup algebras, principal irreducible representations, and characters of commutative semigroups. The book aims to provide a systematic exposition of the algebraic theory of semigroups, building on the foundational work of A. K. Suschkewitsch and D. Rees. It is intended for readers with a background in set theory, mappings, groups, and lattices, and includes exercises to reinforce the concepts discussed. The authors acknowledge the contributions of various scholars and provide a detailed bibliography for further reading.