An ordered regular semigroup, ????, is said to be principally ordered if for every ???? ∈ ???? there exists ???? ∗ = max{???? ∈ ????|???????????? ≤ ????}. A principally ordered regular semigroup is pointed if for every element, ????, we have ???? 2 ≤ ????. Here we investigate those principally ordered regular semigroups that are eventually pointed in the sense that for all ???? ∈ ???? there exists a positive integer, ????, such that (???? ????) 2 ≤ ???? ???? . Necessary and sufficient conditions for an eventually pointed principally ordered regular semigroup to be naturally ordered and to be completely simple are obtained. We describe the subalgebra of (????, ∗ ) generated by a pair of comparable idempotents ???? and ????such that ???? 0 = ???? 0 .

An ordered regular semigroup, ????, is said to be principally ordered if for every ???? ∈ ???? there exists ???? ∗ = max{???? ∈ ????|???????????? ≤ ????}. A principally ordered regular semigroup is pointed if for every element, ????, we have ???? 2 ≤ ????. Here we investigate those principally o...