This paper is a continuation of
a previous paper, in which the structure of certain regular semigroups, called
generalized inverse semigroups, has been studied. A semigroup is called strictly
regular if it is regular and the set of all its idempotents is a subsemigroup. A
generalized inverse semigroup is strictly regular, but the converse is not true. Hence,
the class of generalized inverse semigroups is properly contained in the class of
strictly regular semigroups. The main purpose of this paper is to establish some
results which clarify the structure of strictly regular semigroups. The concept of a
quasi-direct product of a band (that is, an idempotent semigroup) and an inverse
semigroup is introduced, and in particular it is proved that any semigroup is strictly
regular if and only if it is a quasi-direct product of a band and an inverse
semigroup.
