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[1]),[2] or just π-regular semigroup[3] (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.[4]
Epigroups were first studied by Douglas Munn in 1961, who called them pseudoinvertible.[5]
By analogy with periodic semigroups, an epigroup S is partitioned in classes given by its idempotents, which act as identities for each subgroup. For each idempotent e of S, the set:
Ke=\{x\inS\mid\existsn>0:xn\inGe\}
Subsemigroups of an epigroup need not be epigroups, but if they are, then they are called subepigroups. If an epigroup S has a partition in unipotent subepigroups (i.e. each containing a single idempotent), then this partition is unique, and its components are precisely the unipotency classes defined above; such an epigroup is called unipotently partionable. However, not every epigroup has this property. A simple counterexample is the Brandt semigroup with five elements B2 because the unipotency class of its zero element is not a subsemigroup. B2 is actually the quintessential epigroup that is not unipotently partionable. An epigroup is unipotently partionable if and only if it contains no subsemigroup that is an ideal extension of a unipotent epigroup by B2.[5]