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**Fundamentals of semigroup theory.**
*(English)*
Zbl 0835.20077

London Mathematical Society Monographs. New Series. 12. Oxford: Clarendon Press. x, 351 p. (1995).

It is 20 years since the appearance of the author’s first book, An introduction to semigroup theory (1976; Zbl 0355.20056). This new book began as a new edition of the first, but has finished as something more.

A glance at the first five chapters (Introductory Ideas, Green’s Relations, 0-simple Semigroups, Completely Regular Semigroups, Inverse Semigroups) reveals little change, and indeed the now familiar section headings are repeated. However, even in the early chapters there is new material, and the amount of additional content represented by exercises has grown by more than one third overall. For example, the reader will discover a section introducing Sandwich Sets; relational morphisms and the notion of division, central ideas in the modern theory of finite semigroups, are introduced through exercises. The McAlister \(P\)-theory and the related description of free inverse semigroups, very new developments in the mid-70’s, take their deserved place in the revised Chapter 5.

Chapter 6, entitled ‘Other Classes of Regular Semigroups’, gives elementary results on locally inverse, and on orthodox semigroups, while the main result in the final section on semibands is Howie’s own proof that the monoid of singular mappings of the finite full transformation semigroup is idempotent-generated. In this chapter there is a loss of material, and a reader interested in Hall’s analogue of the Munn semigroup for orthodox semigroups should consult the original edition.

Chapter 7 is a new and short account of properties of free semigroups and codes. The author has however resisted the temptation to introduce new chapters on commutative semigroups, and on finite semigroups, important topics which have been treated in other books, (in particular, the recent book Semigroups by P. A. Grillet (1995; Zbl 0830.20079) makes for a nice introduction to the subject at the graduate level, which is somewhat complementary in content to Howie’s account).

The final chapter is, as before, on Semigroup Amalgams and the Zigzag Theorem, and here there is a substantial addition in the form of Renshaw’s research on direct limits and the so-called extension property, which is an analogue of the corresponding results of Cohn in ring theory. This represents the most difficult part of the book, and would be the hardest for the uninitiated to digest, bringing me to a final general comment concerning the level of difficulty of the text. The original book described itself as pitched at the undergraduate/graduate level, while the present work is called a graduate introduction. This perhaps says more about the lower standards expected of undergraduates than about a true change in the nature of the book, but it has freed the author, allowing him to be a little more thorough and uncompromising. For instance, the section on varieties of bands is now preceded by a proof of Birkhoff’s Theorem which was not included in the first edition, presumably to avoid overwhelming the undergraduate reader with a strong dose of universal algebra. Nonetheless the book remains the most suitable on which to base a modern introductory course in algebraic semigroups.

A glance at the first five chapters (Introductory Ideas, Green’s Relations, 0-simple Semigroups, Completely Regular Semigroups, Inverse Semigroups) reveals little change, and indeed the now familiar section headings are repeated. However, even in the early chapters there is new material, and the amount of additional content represented by exercises has grown by more than one third overall. For example, the reader will discover a section introducing Sandwich Sets; relational morphisms and the notion of division, central ideas in the modern theory of finite semigroups, are introduced through exercises. The McAlister \(P\)-theory and the related description of free inverse semigroups, very new developments in the mid-70’s, take their deserved place in the revised Chapter 5.

Chapter 6, entitled ‘Other Classes of Regular Semigroups’, gives elementary results on locally inverse, and on orthodox semigroups, while the main result in the final section on semibands is Howie’s own proof that the monoid of singular mappings of the finite full transformation semigroup is idempotent-generated. In this chapter there is a loss of material, and a reader interested in Hall’s analogue of the Munn semigroup for orthodox semigroups should consult the original edition.

Chapter 7 is a new and short account of properties of free semigroups and codes. The author has however resisted the temptation to introduce new chapters on commutative semigroups, and on finite semigroups, important topics which have been treated in other books, (in particular, the recent book Semigroups by P. A. Grillet (1995; Zbl 0830.20079) makes for a nice introduction to the subject at the graduate level, which is somewhat complementary in content to Howie’s account).

The final chapter is, as before, on Semigroup Amalgams and the Zigzag Theorem, and here there is a substantial addition in the form of Renshaw’s research on direct limits and the so-called extension property, which is an analogue of the corresponding results of Cohn in ring theory. This represents the most difficult part of the book, and would be the hardest for the uninitiated to digest, bringing me to a final general comment concerning the level of difficulty of the text. The original book described itself as pitched at the undergraduate/graduate level, while the present work is called a graduate introduction. This perhaps says more about the lower standards expected of undergraduates than about a true change in the nature of the book, but it has freed the author, allowing him to be a little more thorough and uncompromising. For instance, the section on varieties of bands is now preceded by a proof of Birkhoff’s Theorem which was not included in the first edition, presumably to avoid overwhelming the undergraduate reader with a strong dose of universal algebra. Nonetheless the book remains the most suitable on which to base a modern introductory course in algebraic semigroups.

Reviewer: P.M.Higgins (Colchester)

### MSC:

20Mxx | Semigroups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20M10 | General structure theory for semigroups |

20M17 | Regular semigroups |

20M05 | Free semigroups, generators and relations, word problems |

20M07 | Varieties and pseudovarieties of semigroups |

20M18 | Inverse semigroups |

20M19 | Orthodox semigroups |

20M20 | Semigroups of transformations, relations, partitions, etc. |