In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, 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 (just notation, not necessarily th ...
in which every element has a power that belongs to a
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
. Formally, for all ''x'' in a semigroup ''S'', there exists a positive
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
''n'' and a subgroup ''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. 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
Class, Classes, or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used d ...
of epigroups also contains all
completely regular semigroups and all
completely 0-simple semigroups.
* All epigroups are also
eventually regular semigroups. (also known as π-regular semigroups)
* A
cancellative epigroup is a
group.
*
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
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 (just notation, not necessarily the ...
of ''S'' is also an epigroup.
* In an epigroup the
Nambooripad order (as extended by P.R. Jones) and the
natural partial order (of Mitsch) coincide.
Examples
* The semigroup of all
square matrices of a given size over a
division ring
In algebra, a division ring, also called a skew field (or, occasionally, a sfield), 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 multiplicativ ...
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:
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 ''B
2'' because the unipotency class of its zero element is not a subsemigroup. ''B
2'' 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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it contains no subsemigroup that is an
ideal extension of a unipotent epigroup by ''B
2''.
See also
Special classes of semigroups
References
{{reflist
Semigroup theory
Algebraic structures