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: [math]\displaystyle{ K_e = \{ x \in S \mid \exists n\gt 0 : x^n \in G_e \} }[/math] 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
References
- ↑ 1.0 1.1 Lex E. Renner (2005). Linear Algebraic Monoids. Springer. pp. 27–28. ISBN 978-3-540-24241-3. https://books.google.com/books?id=VSEce2_LJ20C&pg=PA27.
- ↑ A. V. Kelarev, Applications of epigroups to graded ring theory, Semigroup Forum, Volume 50, Number 1 (1995), 327–350 doi:10.1007/BF02573530
- ↑ Eric Jespers; Jan Okninski (2007). Noetherian Semigroup Algebras. Springer. p. 16. ISBN 978-1-4020-5809-7. https://books.google.com/books?id=qzj3MkWnCLQC&pg=PA16.
- ↑ 4.0 4.1 Andrei V. Kelarev (2002). Ring Constructions and Applications. World Scientific. ISBN 978-981-02-4745-4.
- ↑ 5.0 5.1 5.2 5.3 5.4 Lev N. Shevrin (2002). "Epigroups". in Aleksandr Vasilʹevich Mikhalev and Günter Pilz. The Concise Handbook of Algebra. Springer. pp. 23–26. ISBN 978-0-7923-7072-7. https://books.google.com/books?id=i2g2cstPDfEC&pg=PA24.
- ↑ Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. p. 4. ISBN 978-0-19-853577-5.
- ↑ Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. p. 50. ISBN 978-0-19-853577-5.
- ↑ Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. p. 12. ISBN 978-0-19-853577-5.
- ↑ Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. p. 28. ISBN 978-0-19-853577-5.
- ↑ Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press. p. 48. ISBN 978-0-19-853577-5.
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