Right P-Comparable Semigroups

In this article we utilise the notion of right waist and right comparizer to study the ideal theory of semigroups. We also consider which of the properties of right cones can be carried over to right P-comparable semigroups. We give sufficient and necessary conditions on the set of nilpotent elements of a semigroup to be an ideal, and we provide several equivalent characterizations for a right ideal being a right waist. In one of our main results we show that in a right P ₁-comparable semigroup with left cancellation law, a prime segment P ₂ ? P ₁ is Archimedean, simple or exceptional. This extends a similar result pertaining to right cones.