Geodesic
Ordered Regular Semigroups with Biggest Associates
Discussiones Mathematicae. General Algebra and Applications, Tome 39 (2019) no. 1, pp. 5-21

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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 xo. 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 
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Blyth, T.S.; Almeida Santos, M.H. Ordered Regular Semigroups with Biggest Associates. Discussiones Mathematicae. General Algebra and Applications, Tome 39 (2019) no. 1, pp. 5-21. http://geodesic.mathdoc.fr/item/DMGAA_2019_39_1_a0/