Geodesic
Pointed principally ordered regular semigroups
Discussiones Mathematicae. General Algebra and Applications, Tome 36 (2016) no. 1, pp. 101-111

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An ordered semigroup S is said to be principally ordered if, for every x ∈ S there exists x* = maxy ∈ S | xyx ⩽ x. Here we investigate those principally ordered regular semigroups that are pointed in the sense that the classes modulo Green's relations ℒ,ℛ, have biggest elements which are idempotent. Such a semigroup is necessarily a semiband. In particular we describe the subalgebra of (S;*) generated by a pair of comparable idempotents that are -related. We also prove that those -classes which are subsemigroups are ordered rectangular bands.
Keywords: regular semigroup, principally ordered, naturally ordered, Green's relations 
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Blyth, T.; Pinto, G. Pointed principally ordered regular semigroups. Discussiones Mathematicae. General Algebra and Applications, Tome 36 (2016) no. 1, pp. 101-111. http://geodesic.mathdoc.fr/item/DMGAA_2016_36_1_a7/