Pointed principally ordered regular semigroups

An ordered semigroup S is said to be principally ordered if, for every x ∈ S there exists x* = max{y ∈ 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.