Semigroups with Modular or Semimodular Subsemigroup Lattices

Abstract

Restrictions imposed on subsemigroup lattices may be of different kinds. One of the prevalent types of restrictions is that of a lattice satisfying a fixed system of identities, in particular, a fixed identity. Classical examples are presented by distributivity and modularity. Semigroups with distributive and modular subsemigroup lattices were described in the beginning of investigations on lattice properties of semigroups (in the modular case it was done modulo groups). The similarity of the resulting descriptions (which differ in restrictions for maximal subgroups only) provoked a question about possible unification of these results. It turns out that such unification can be formulated in terms of an arbitrary modular variety of lattices (i.e. any variety consisting of modular lattices). This result is central in Section 6. (Note that this topic is continued later in the book; namely, Section 29 is devoted to considerations of semigroups whose subsemigroup lattices belong to a non-trivial variety; see also II.2.) This result is based on the more general considerations of Section 5, where semigroups with semimodular subsemigroup lattices are described (modulo groups). A very special case of distributivity is the property of being a chain. We call a semigroup whose subsemigroups form a chain a chain semigroup. We close Section 6 with a description of chain semigroups.