Embedding Semigroups with Associate Subgroups into Semidirect Products

Abstract

A semigroup S is said to have an associate subgroup G if, for each s ∈ S, there is a unique s* ∈ G such that ss*s = s. If the identity 1 _G of G is medial, i.e., c1 _G c = c holds for each c being a product of idempotents, we show that S is isomorphic to a certain subsemigroup of a semidirect product of an idempotent generated semigroup C by G. If additionally S is orthodox, we may choose C to be a band, belonging to the band variety, generated by the band of idempotents of S.

Key Words:

• Associate subgroup
• Idempotent generated semigroup
• Medial idempotent
• Semidirect product

2000 Mathematics Subject Classification:

• 20M17
• 20M19

Notes

Communicated by V. Gould.