Residuated Completely Simple Semigroups
T. S. Blyth, Gama Pinto · Algebra Colloquium · 2014
We consider particular compatible orders on a given completely simple semigroup Sx=M(〈x〉; I ,Λ;P) where 〈x〉 is an ordered cyclic group with x > 1 and P11=x-1. Of these, only the lexicographic and bootlace orders yield residuated semigroups. With the lexicographic order, Sx is orthodox and has a biggest idempotent. With the bootlace order, the maximal idempotents of Sx are identified by specific locations in the sandwich matrix. In the orthodox case there is also a biggest idempotent and, for sandwich matrices of a given size, uniqueness up to ordered semigroup isomorphism is established.