E-semigroup

In the area of mathematics known as semigroup theory, an E-semigroup is a semigroup in which the idempotents form a subsemigroup.^[1]

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.^[1]

References

1. Weipoltshammer, B. (2002). "Certain congruences on E-inversive E-semigroups". Semigroup Forum 65 (2): 233. doi:10.1007/s002330010131.