# Epigroup

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{{Short description|Type of semigroup}}
In [abstract algebra](/source/abstract_algebra), an '''epigroup''' is a [semigroup](/source/semigroup) in which every element has a power that belongs to a [subgroup](/source/subgroup). Formally, for all ''x'' in a semigroup ''S'', there exists a positive [integer](/source/integer) ''n'' and a subgroup ''G'' of ''S'' such that ''x''<sup>''n'' </sup> belongs to&nbsp;''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](/source/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](/source/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](/source/Walter_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 semigroup](/source/periodic_semigroup)s,<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 semigroup](/source/finite_semigroup)s are also epigroups.
* The [class](/source/class_(set_theory)) of epigroups also contains all [completely regular semigroup](/source/completely_regular_semigroup)s and all [completely 0-simple semigroup](/source/completely_0-simple_semigroup)s.<ref name="MikhalevPilz2002"/>
* All epigroups are also [eventually regular semigroup](/source/eventually_regular_semigroup)s.<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](/source/cancellative_semigroup) epigroup is a [group](/source/group_(mathematics)).<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](/source/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](/source/regular_semigroup) [subsemigroup](/source/Semigroup) of ''S'' is also an epigroup.<ref name="Renner2005"/>
* In an epigroup the [Nambooripad order](/source/Nambooripad_order) (as extended by P.R. Jones) and the [natural partial order](/source/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](/source/square_matrices) of a given size over a [division ring](/source/division_ring) is an epigroup.<ref name="MikhalevPilz2002"/>
* The multiplicative semigroup of every [semisimple Artinian ring](/source/semisimple_Artinian_ring) is an epigroup.<ref name="Kelarev2002"/>{{rp|5}}
* Any [algebraic semigroup](/source/algebraic_semigroup) is an epigroup.

== Structure ==
By analogy with periodic semigroups, an epigroup ''S'' is [partitioned](/source/Partition_of_a_set) in classes given by its [idempotent](/source/idempotent)s, 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](/source/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](/source/if_and_only_if) it contains no subsemigroup that is an [ideal extension](/source/ideal_extension) of a unipotent epigroup by ''B<sub>2</sub>''.<ref name="MikhalevPilz2002"/>

==See also==
[Special classes of semigroups](/source/Special_classes_of_semigroups)

== References ==
{{reflist}}

Category:Semigroup theory
Category:Algebraic structures

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