  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.
