E-semigroup

In the area of mathematics known as semigroup theory, an E-semigroup is a semigroup in which the idempotents form a subsemigroup.
Certain classes of E-semigroups have been studied long before the more general class, in particular, a regular semigroup that is also an E-semigroup is known as an orthodox semigroup.
Weipoltshammer proved that the notion of weak inverse can also be used to define/characterize E-semigroups as follows: a semigroup S is an E-semigroup if and only if, for all a and b ∈ S, W = W''W'', where W ≝ is the set of weak inverses of a.