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.
