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 (the existence of which is one way to define "E"-inversive semigroups) 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"("ab") = "W"("b")"W"("a"), where "W"("x") ≝ {"x" ∈ "S" | "xax" = "x"} is the set of weak inverses of "x".