Semigroups and their subsemigroup lattices.

*(English)*Zbl 0858.20054
Mathematics and its Applications (Dordrecht) 379. Dordrecht: Kluwer Academic Publishers (ISBN 0-7923-4221-6). xi, 378 p. (1996).

The study of the subalgebra lattices of algebras (of a given type) began in the realm of group theory, the current state of affairs there being the topic of the recent monograph by R. Schmidt [Subgroup Lattices of Groups (Walter de Gruyter, Berlin 1994; Zbl 0843.20003)]. The first papers on the lattice of subsemigroups of a semigroup were published in 1951. The monograph under review is a comprehensive survey of the field, an area dominated by Russian researchers and, in particular, by the first author himself. It is a translation and nontrivial expansion of the Russian original [Semigroups and their subsemigroup lattices, Ural University, Sverdlovsk (Part 1, 1990; Zbl 0760.20016, Part 2, 1991; Zbl 0760.20017)], itself based on the survey article [Semigroup Forum 27, 1-154 (1983; Zbl 0523.20037)].

The primary topic of the book is the lattice \(\text{Sub }S\) of subsemigroups (including the empty one) of a semigroup \(S\). The authors identify three main aspects, each of which we illustrate with some sample results: A) Restrictions on subsemigroup lattices: for instance finiteness of \(\text{Sub }S\) implies finiteness of \(S\); much less clear are characterizations of those semigroups for which \(\text{Sub }S\) is semimodular, modular or distributive (§6); B) Properties of subsemigroup lattices: for instance (§25) the class of bands (= idempotent semigroups) is “lattice-elementary”, in the sense that the class of subsemigroup lattices of bands is described by a set of lattice-theoretic sentences, within the class of all subsemigroup lattices; (§28) the finite lattices embeddable in the subsemigroup lattices of free semigroups consist of the finite “lower bounded” lattices; C) Lattice isomorphisms: for instance (§35), the class of commutative Archimedean semigroups is lattice closed in the class of commutative semigroups, in the sense that any commutative semigroup whose subsemigroup lattice is isomorphic to that of an Archimedean commutative semigroup is also Archimedean; any free semigroup \(F\) is strictly lattice determined, in the sense that any isomorphism of \(\text{Sub }F\) on \(\text{Sub }S\), for a semigroup \(S\), is induced by an isomorphism or anti-isomorphism between \(F\) and \(S\) (§34).

Of necessity, many questions about subsemigroup lattices reduce to the corresponding questions about the subgroup lattices of a related class of groups. More generally, unary semigroups (semigroups with an additional unary operation) play a role throughout the book. In particular, epigroups – semigroups in which some power of each element lies in a subgroup – arise frequently. Many results on their subsemigroup lattices lead to parallel considerations of their lattices of subepigroups; recent investigations have focused on the lattices of completely regular subsemigroups of completely regular semigroups, for instance. Similarly, extensive coverage is given to inverse semigroups and their lattices of inverse subsemigroups. Their lattices of full inverse subsemigroups (those containing all the idempotents) also receive attention.

There are copious, informative notes on each chapter, including references to much additional material; there are also exercises for each chapter. The book is extremely well organized and, despite the authors’ modest disclaimer, the English is in general excellent. The overall scope of the book can be gauged by the list of topics, subdivided according to the three aspects described above: Part A. Semigroups with Certain Types of Subsemigroup Lattices. (I) Preliminaries, (II) Semigroups with Modular or Semimodular Subsemigroup Lattices, (III) Semigroups with Complementable Subsemigroups, (IV) Finiteness Conditions, (V) Inverse Semigroups with Certain Types of Lattices of Inverse Subsemigroups, (VI) Inverse Semigroups with Certain Types of Lattices of Full Inverse Subsemigroups. Part B. Properties of Subsemigroup Lattices. (VII) Lattice Characteristics of Classes of Semigroups, (VIII) Embedding Lattices in Subsemigroup Lattices. Part C. Lattice Isomorphisms. (IX) Preliminaries on Lattice Isomorphisms, (X) Cancellative Semigroups, (XI) Commutative Semigroups, (XII) Semigroups Decomposable into Rectangular Bands, (XIII) Semigroups Defined by Certain Presentations, (XIV) Inverse Semigroups.

The primary topic of the book is the lattice \(\text{Sub }S\) of subsemigroups (including the empty one) of a semigroup \(S\). The authors identify three main aspects, each of which we illustrate with some sample results: A) Restrictions on subsemigroup lattices: for instance finiteness of \(\text{Sub }S\) implies finiteness of \(S\); much less clear are characterizations of those semigroups for which \(\text{Sub }S\) is semimodular, modular or distributive (§6); B) Properties of subsemigroup lattices: for instance (§25) the class of bands (= idempotent semigroups) is “lattice-elementary”, in the sense that the class of subsemigroup lattices of bands is described by a set of lattice-theoretic sentences, within the class of all subsemigroup lattices; (§28) the finite lattices embeddable in the subsemigroup lattices of free semigroups consist of the finite “lower bounded” lattices; C) Lattice isomorphisms: for instance (§35), the class of commutative Archimedean semigroups is lattice closed in the class of commutative semigroups, in the sense that any commutative semigroup whose subsemigroup lattice is isomorphic to that of an Archimedean commutative semigroup is also Archimedean; any free semigroup \(F\) is strictly lattice determined, in the sense that any isomorphism of \(\text{Sub }F\) on \(\text{Sub }S\), for a semigroup \(S\), is induced by an isomorphism or anti-isomorphism between \(F\) and \(S\) (§34).

Of necessity, many questions about subsemigroup lattices reduce to the corresponding questions about the subgroup lattices of a related class of groups. More generally, unary semigroups (semigroups with an additional unary operation) play a role throughout the book. In particular, epigroups – semigroups in which some power of each element lies in a subgroup – arise frequently. Many results on their subsemigroup lattices lead to parallel considerations of their lattices of subepigroups; recent investigations have focused on the lattices of completely regular subsemigroups of completely regular semigroups, for instance. Similarly, extensive coverage is given to inverse semigroups and their lattices of inverse subsemigroups. Their lattices of full inverse subsemigroups (those containing all the idempotents) also receive attention.

There are copious, informative notes on each chapter, including references to much additional material; there are also exercises for each chapter. The book is extremely well organized and, despite the authors’ modest disclaimer, the English is in general excellent. The overall scope of the book can be gauged by the list of topics, subdivided according to the three aspects described above: Part A. Semigroups with Certain Types of Subsemigroup Lattices. (I) Preliminaries, (II) Semigroups with Modular or Semimodular Subsemigroup Lattices, (III) Semigroups with Complementable Subsemigroups, (IV) Finiteness Conditions, (V) Inverse Semigroups with Certain Types of Lattices of Inverse Subsemigroups, (VI) Inverse Semigroups with Certain Types of Lattices of Full Inverse Subsemigroups. Part B. Properties of Subsemigroup Lattices. (VII) Lattice Characteristics of Classes of Semigroups, (VIII) Embedding Lattices in Subsemigroup Lattices. Part C. Lattice Isomorphisms. (IX) Preliminaries on Lattice Isomorphisms, (X) Cancellative Semigroups, (XI) Commutative Semigroups, (XII) Semigroups Decomposable into Rectangular Bands, (XIII) Semigroups Defined by Certain Presentations, (XIV) Inverse Semigroups.

Reviewer: P.R.Jones (Milwaukee)

##### MSC:

20M10 | General structure theory for semigroups |

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

08A30 | Subalgebras, congruence relations |

20E15 | Chains and lattices of subgroups, subnormal subgroups |

06B15 | Representation theory of lattices |