Semigroups Defined by Certain Presentations

Abstract

Like any algebraic system, every semigroup can be defined by a presentation, i.e. by a system of defining relations over some generating set. It is reasonable to be interested in the following quetion: how do lattice isomorphisms depend on presentations? As to the problem of lattice classification, it is scarcely possible to find rather general approaches from this viewpoint, although for a number of classes of semigroups (for example, for cyclic semigroups, see XIII.4), one has succeeded in obtaining an exhaustive solution of this problem in terms of presentations. Particular attention was paid here to discovering various types of presentations which ensure lattice determinability of the respective semigroups. Fairly often such presentations are characterized by some “freedom”. In the extreme case of such freedom, i.e. when there are no non-trivial relations, we deal simply with free semigroups; their strict lattice determinability has already been proved in Section 33 (see 33.29) as a consequence of more general facts on lattice isomorphisms of cancellative semigroups. (Notice that this result, in its turn, was apparently the first one in investigations of subsemigroup lattices of semigroups per se; see X.1). But the property of being a free semigroup can be essentially generalized in another direction, by going over to semigroups which are decomposable into a free product; from the viewpoint of presentations, the last property means that it is possible to divide the set of relations into subsets (more than one) such that generators which occur in relations of some such subset do not occur in the relations of the other subsets. Section 40 is devoted to the proof of strict lattice determinability of semigroups decomposable into a free product. This result has an independent significance which is emphasized, in particular, by Corollary 40.28; besides this fact, it is applied, together with the main result of Section 39, in Section 41 to the study of lattice isomorphisms of finitely presented semigroups. The central result in Section 41 establishes the lattice determinability of semigroups whose presentations have the number of generators, which is considerably greater than that of the defining relations (see Theorem 41.5); it is of interest to note that the peculiarities of the form of the relations play no role here. Another result of this section (Theorem 41.8) states that a bicyclic semigroup is strictly lattice-determined; as is well known, this semigroup can be presented (in the class of all semigroups) by two generators and four relations.