Epigroup Explained

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

Examples

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:

Ke=\{x\inS\mid\existsn>0:xn\inGe\}

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

Notes and References

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