Ordered Regular Semigroups with Biggest Associates

Abstract

We investigate the class BA of ordered regular semigroups in which each element has a biggest associate x† = max {y | xyx = x}. This class properly contains the class PO of principally ordered regular semigroups (in which there exists x⋆ = max {y | xyx ⩽x}) and is properly contained in the class BI of ordered regular semigroups in which each element has a biggest inverse x◦. We show that several basic properties of the unary operation x ↦ x⋆ in PO extend to corresponding properties of the unary operation x ↦ x† in BA. We consider naturally ordered semigroups in BA and prove that those that are orthodox contain a biggest idempotent. We determine the structure of some such semigroups in terms of a principal left ideal and a principal right ideal. We also characterise the completely simple members of BA. Finally, we consider the naturally ordered semigroups in BA that do not have a biggest idempotent.

Keywords

• regular semigroup
• biggest associate
• principally ordered
• naturally ordered
• 20m20
• 20m17