What is "epigroup"

In abstract algebra, an epigroup is a semigroup in which every element has a power that belongs to a subgroup. Formally, for all x in a semigroup S, there exists a positive integer n and a subgroup G of S such that x belongs to G.

Epigroups are known by wide variety of other names, including quasi-periodic semigroup, group-bound semigroup, completely π-regular semigroup, strongly π-regular semigroup (sπr), or just π-regular semigroup (although the latter is ambiguous).

More generally, in an arbitrary semigroup an element is called group-bound if it has a power that belongs to a subgroup.

Epigroups have applications to ring theory. Many of their properties are studied in this context.

Epigroups were first studied by Douglas Munn in 1961, who called them pseudoinvertible.