Epigroup

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]

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]

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

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: ${\displaystyle K_{e}=\{x\in S\mid \exists n>0:x^{n}\in G_{e}\}}$ 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 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 it contains no subsemigroup that is an ideal extension of a unipotent epigroup by B₂.^[5]

See also

Special classes of semigroups

References

1. Lex E. Renner (2005). Linear Algebraic Monoids. Springer. pp. 27–28. ISBN 978-3-540-24241-3.
2. A. V. Kelarev, Applications of epigroups to graded ring theory, Semigroup Forum, Volume 50, Number 1 (1995), 327–350 doi:10.1007/BF02573530
3. Eric Jespers; Jan Okninski (2007). Noetherian Semigroup Algebras. Springer. p. 16. ISBN 978-1-4020-5809-7.
4. Andrei V. Kelarev (2002). Ring Constructions and Applications. World Scientific. ISBN 978-981-02-4745-4.
5. Lev N. Shevrin (2002). "Epigroups". In Aleksandr Vasilʹevich Mikhalev and Günter Pilz (ed.). The Concise Handbook of Algebra. Springer. pp. 23–26. ISBN 978-0-7923-7072-7.
6. Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. p. 4. ISBN 978-0-19-853577-5.
7. Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. p. 50. ISBN 978-0-19-853577-5.
8. Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. p. 12. ISBN 978-0-19-853577-5.
9. Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. p. 28. ISBN 978-0-19-853577-5.
10. Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. p. 48. ISBN 978-0-19-853577-5.