Subdirectly irreducible idempotent semigroups

Abstract

Subdirectly irreducible idempotent semigroups have been discussed by B.M. Schein. Using his results, a subdirectly irreducible idempotent semigroup is shown to be either a semigroup of mappings of a set into itself, with certain stated conditions, or the dual of such a semigroup, or one of these with an adjoined zero. This characterization is used to show that, with a few exceptions, the join reducible elements of the lattice of equational classes of idempotent semigroups contain only subdirectly irreducible members belonging to proper subclasses. This result gives structure theorems, special cases of which appear in the literature. An example is also given of an infinite subdirectly irreducible idempotent semigroup in the equational class of idempotent semigroups defined by xyx = xy.