Epigroup

Properties

• Epigroups are a generalization of periodic semigroups, thus all finite semigroups are also epigroups.
• The class of epigroups also contains all completely regular semigroups and all completely 0-simple semigroups.
• All epigroups are also eventually regular semigroups. (also known as π-regular semigroups)
• A cancellative epigroup is a group.
• Green's relations D and J coincide for any epigroup.
• If S is an epigroup, any regular subsemigroup of S is also an epigroup.
• In an epigroup the Nambooripad order (as extended by P.R. Jones) and the natural partial order (of Mitsch) coincide.

Examples

• The semigroup of all square matrices of a given size over a division ring is an epigroup.
• The multiplicative semigroup of every semisimple Artinian ring is an epigroup.
• 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: K_e = { x \in S \mid \exists n0 : x^n \in G_e } is called a unipotency class (whereas for periodic semigroups the usual name is torsion class.)

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 B2 because the unipotency class of its zero element is not a subsemigroup. B2 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 B2.

References

References

1. Lex E. Renner. (2005). "Linear Algebraic Monoids". Springer.
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. (2007). "Noetherian Semigroup Algebras". Springer.
4. Andrei V. Kelarev. (2002). "Ring Constructions and Applications". World Scientific.
5. Lev N. Shevrin. (2002). "The Concise Handbook of Algebra". Springer.
6. Peter M. Higgins. (1992). "Techniques of semigroup theory". Oxford University Press.
7. Peter M. Higgins. (1992). "Techniques of semigroup theory". Oxford University Press.
8. Peter M. Higgins. (1992). "Techniques of semigroup theory". Oxford University Press.
9. Peter M. Higgins. (1992). "Techniques of semigroup theory". Oxford University Press.
10. Peter M. Higgins. (1992). "Techniques of semigroup theory". Oxford University Press.