Unary Semigroups with an Associate Subgroup

Abstract

A subgroup H of a regular semigroup S is said to be an associate subgroup of S if for every s ∈ S, there is a unique associate of s in H. An idempotent z of S is said to be medial if czc = c, for every c product of idempotents of S. Blyth and Martins established a structure theorem for semigroups with an associate subgroup whose identity is a medial idempotent, in terms of an idempotent generated semigroup, a group and a single homomorphism. Here, we construct a system of axioms which characterize these semigroups in terms of a unary operation satisfying those axioms. As a generalization of this class of semigroups, we characterize regular semigroups S having a subgroup which is a transversal of a congruence on S.

Key Words:

• Associate
• Medial
• Semigroup
• Subgroup
• Unary

Mathematics Subject Classification:

• Primary 20M10

ACKNOWLEDGMENT

We are indebted to the referee for careful reading of the article.

This work was partly supported by F.C.T. through the Centro de Matemática da Universidade do Minho, Braga, Portugal.

Notes

Communicated by D. Easdown.

Additional information

Notes on contributors

Mario Petrich,*

*Visiting professor