Residuated completely simple semigroups

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 S_x 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.