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 xn 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 in 1961, who called them pseudoinvertible. (Wikipedia).