Right $P$-comparable semigroups

In this paper we introduce the notion of right waist and right comparizer ideals for semigroups. In particular, we study the ideal theory of semigroups containing right waists and right comparizer ideals. We also study those properties of right cones that 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. We provide several equivalent characterizations for a right ideal being a right waist. In one of our main result we show that in a right $P_1$-comparable semigroup with left cancellation law, a prime segment $P_2\subset P_1$ is Archimedean, simple or exceptional, extending a similar result of right cones to $P$-comparable semigroups.