Group-bound semigroup

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 Douglas Munn in 1961, who called them pseudoinvertible.

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: ${\displaystyle K_{e}=\{x\in S\;|\;\exists n>0:\;x^{n}\in G_{e}\}}$ is called a unipotency class (whereas for periodic semigroups the usual name is torsion class.)

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