An extension of the Rådström cancellation theorem to cornets

Abstract

The aim of this paper is to introduce the notion of cornets, which form a particular subclass of ordered semigroups also equipped with a multiplication by natural numbers. The most important standard examples for cornets are the families of the nonempty subsets and the nonempty fuzzy subsets of a vector space. In a cornet, the convexity, nonnegativity, Archimedean property, boundedness, closedness of an element can be defined naturally. The basic properties related to these notions are established. The main result extends the Cancellation Principle discovered by Rådström in 1952.

Order Cancellation Law in a Semigroup of Closed Convex Sets

“…The order cancellation law or Rådström cancellation theorem [23,25], investigated also in [15,18,20], enables an embedding of a semigroup of convex sets into a quotient vector space called a Minkowski-Rådström-Hörmander space [12,28]. This space plays a crucial role in differentiation of nonsmooth functions (quasidifferential calculus of Demyanov and Rubinov [1,7,8,11,14,21]) and in integration and differentiation in the theory of multifunctions [5,6].…”

Order cancellation law in the family of bounded convex sets

“…The order cancellation law or Rådström cancellation theorem [22,24], investigated also in [14,17,19], enables an embedding of a semigroup of convex sets into a quotient vector space called a Minkowski-Rådström-Hörmander space [11,27]. This space plays a crucial role in differentiation of nonsmooth functions (quasidifferential calculus of Demyanov and Rubinov [1,6,7,10,13,20]) and in integration and differentiation in the theory of multifunctions [4,5].…”