abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...

, an epigroup is a

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

in which every element has a power that belongs to a

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

. Formally, for all ''x'' in a semigroup ''S'', there exists a positive integer ''n'' and a

''G'' of ''S'' such that ''x''^''n'' 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 In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...

. Many of their properties are studied in this context. Epigroups were first studied by Douglas Munn in 1961, who called them ''pseudoinvertible''.

Properties

* Epigroups are a generalization of periodic semigroups, thus all finite semigroups are also epigroups. * The class of epigroups also contains all completely regular semigroups and all completely 0-simple semigroups. * All epigroups are also

eventually regular semigroup In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consis ...

s. (also known as π-regular semigroups) * A cancellative epigroup is a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

. *

Green's relations In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951 ...

''D'' and ''J'' coincide for any epigroup. * If ''S'' is an epigroup, any regular subsemigroup of ''S'' is also an epigroup. * In an epigroup the

Nambooripad order In mathematics, Nambooripad order (also called Nambooripad's partial order) is a certain natural partial order on a regular semigroup discovered by K S S Nambooripad in late seventies. Since the same partial order was also independently discovere ...

(as extended by P.R. Jones) and the natural partial order (of Mitsch) coincide.

Examples

* The semigroup of all matrices over a

division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...

is an epigroup. * The multiplicative semigroup of every semisimple Artinian ring is an epigroup. * Any algebraic semigroup is an epigroup.

Structure

By analogy with periodic semigroups, an epigroup ''S'' is partitioned in classes given by its

idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

s, which act as identities for each subgroup. For each idempotent ''e'' of ''S'', the set:

K_e= \

is called a ''unipotency class'' (whereas for periodic semigroups the usual name is torsion class.) 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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...

it contains no subsemigroup that is an ideal extension of a unipotent epigroup by ''B₂''.

References

{{reflist Semigroup theory Algebraic structures

Properties

Examples

Structure

See also

References