Ordered semigroups containing maximal or minimal elements (Q1100222)

In this paper we initiate the study of totally ordered semigroups containing maximal or minimal ideals. We shall describe the algebraic nature of maximal and minimal elements; how their presence generates convex ideals and establish the existence of the kernel in case the semigroup does not contain proper convex ideals. We provide also the structure theorems on one-sided cancellative and algebraically simple totally ordered semigroups containing maximal or minimal elements.