Minkowski difference and Sallee elements in an ordered semigroup

In the manner of Pallaschke and Urbański ([5], chapter 3) we generalize the notions of the Minkowski difference and Sallee sets to a semigroup. Sallee set (see [7], definition of the family $S$ on p. 2) is a compact convex subset $A$ of a topological vector space $X$ such that for all subsets $B$ the Minkowski difference $A-B$ of the sets $A$ and $B$ is a summand of $A$. The family of Sallee sets characterizes the Minkowski subtraction, which is important to the arithmetic of compact convex sets (see [5]). Sallee polytopes are related to monotypic polytopes (see [4]). We generalize properties of Minkowski difference and Sallee sets to semigroup and investigate the families of Sallee elements in several specific semigroups.