{{Short description|Type of 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 ''x''<sup>''n'' </sup> 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'''<ref name="Renner2005">{{cite book|author=Lex E. Renner|title=Linear Algebraic Monoids|url=https://books.google.com/books?id=VSEce2_LJ20C&pg=PA27|year=2005|publisher=Springer|isbn=978-3-540-24241-3|pages=27–28}}</ref>),<ref>A. V. Kelarev, ''Applications of epigroups to graded ring theory'', Semigroup Forum, Volume 50, Number 1 (1995), 327–350 {{doi|10.1007/BF02573530}}</ref> or just '''π-regular semigroup'''<ref name="JespersOkninski2007">{{cite book|author1=Eric Jespers|author2=Jan Okninski|title=Noetherian Semigroup Algebras|url=https://books.google.com/books?id=qzj3MkWnCLQC&pg=PA16|year=2007|publisher=Springer|isbn=978-1-4020-5809-7|page=16}}</ref> (although the latter is ambiguous).<!--I've bolded those which seem most common-->
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.<ref name="Kelarev2002">{{cite book|author=Andrei V. Kelarev|title=Ring Constructions and Applications|year=2002|publisher=World Scientific|isbn=978-981-02-4745-4}}</ref>
Epigroups were first studied by Douglas Munn in 1961, who called them ''pseudoinvertible''.<ref name="MikhalevPilz2002">{{cite book|editor=Aleksandr Vasilʹevich Mikhalev and Günter Pilz|title=The Concise Handbook of Algebra|url=https://books.google.com/books?id=i2g2cstPDfEC&pg=PA24|year=2002|publisher=Springer|isbn=978-0-7923-7072-7|pages=23–26|author=Lev N. Shevrin|chapter=Epigroups}}</ref>
== Properties == * Epigroups are a generalization of periodic semigroups,<ref>{{cite book|author=Peter M. Higgins|title=Techniques of semigroup theory|year=1992|publisher=Oxford University Press|isbn=978-0-19-853577-5|page=4}}</ref> thus all finite semigroups are also epigroups. * The class of epigroups also contains all completely regular semigroups and all completely 0-simple semigroups.<ref name="MikhalevPilz2002"/> * All epigroups are also eventually regular semigroups.<ref>{{cite book|author=Peter M. Higgins|title=Techniques of semigroup theory|year=1992|publisher=Oxford University Press|isbn=978-0-19-853577-5|page=50}}</ref> (also known as π-regular semigroups) * A cancellative epigroup is a group.<ref>{{cite book|author=Peter M. Higgins|title=Techniques of semigroup theory|year=1992|publisher=Oxford University Press|isbn=978-0-19-853577-5|page=12}}</ref> * Green's relations ''D'' and ''J'' coincide for any epigroup.<ref>{{cite book|author=Peter M. Higgins|title=Techniques of semigroup theory|year=1992|publisher=Oxford University Press|isbn=978-0-19-853577-5|page=28}}</ref> * If ''S'' is an epigroup, any regular subsemigroup of ''S'' is also an epigroup.<ref name="Renner2005"/> * In an epigroup the Nambooripad order (as extended by P.R. Jones) and the natural partial order (of Mitsch) coincide.<ref>{{cite book|author=Peter M. Higgins|title=Techniques of semigroup theory|year=1992|publisher=Oxford University Press|isbn=978-0-19-853577-5|page=48}}</ref>
== Examples == * The semigroup of all square matrices of a given size over a division ring is an epigroup.<ref name="MikhalevPilz2002"/> * The multiplicative semigroup of every semisimple Artinian ring is an epigroup.<ref name="Kelarev2002"/>{{rp|5}} * Any algebraic semigroup is an epigroup.
== Structure == 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: <math>K_e = \{ x \in S \mid \exists n>0 : x^n \in G_e \}</math> is called a ''unipotency class'' (whereas for periodic semigroups the usual name is torsion class.)<ref name="MikhalevPilz2002"/>
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 ''B<sub>2</sub>'' because the unipotency class of its zero element is not a subsemigroup. ''B<sub>2</sub>'' 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 ''B<sub>2</sub>''.<ref name="MikhalevPilz2002"/>
==See also== Special classes of semigroups
== References == {{reflist}}
Category:Semigroup theory Category:Algebraic structures