Top Qs
Timeline
Chat
Perspective
Epigroup
From Wikipedia, the free encyclopedia
Remove ads
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]
Remove ads
Properties
- Epigroups are a generalization of periodic semigroups,[6] thus all finite semigroups are also epigroups.
- The class of epigroups also contains all completely regular semigroups and all completely 0-simple semigroups.[5]
- All epigroups are also eventually regular semigroups.[7] (also known as π-regular semigroups)
- A cancellative epigroup is a group.[8]
- Green's relations D and J coincide for any epigroup.[9]
- If S is an epigroup, any regular subsemigroup of S is also an epigroup.[1]
- In an epigroup the Nambooripad order (as extended by P.R. Jones) and the natural partial order (of Mitsch) coincide.[10]
Remove ads
Examples
- The semigroup of all square matrices of a given size over a division ring is an epigroup.[5]
- The multiplicative semigroup of every semisimple Artinian ring is an epigroup.[4]: 5
- Any algebraic semigroup is an epigroup.
Structure
Summarize
Perspective
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: is called a unipotency class (whereas for periodic semigroups the usual name is torsion class.)[5]
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 partitionable. 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 partitionable. An epigroup is unipotently partitionable if and only if it contains no subsemigroup that is an ideal extension of a unipotent epigroup by B2.[5]
See also
References
Wikiwand - on
Seamless Wikipedia browsing. On steroids.
Remove ads