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.