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1 Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

1 2007

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Minkowski difference and Sallee elements in an ordered semigroup

Borowska D., Grzybowski J.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2007

Vol. 47, [Z] 1

77-83

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 monotypie poly topes (see [4]). We generalize properties of Minkowski difference and Sallee sets to semigroup and investigate the families of Sallee elements in several specific semigroups.