An extension of the R{\aa}dstr\"om Cancellation Theorem to Cornets

TL;DR

This paper introduces cornets, a new subclass of ordered semigroups with natural number multiplication, and extends R{ a}dstr"om's Cancellation Theorem to this broader context.

Contribution

It defines the concept of cornets and generalizes the R{ a}dstr"om Cancellation Theorem to this new mathematical structure.

Findings

Cornets include families of nonempty subsets and fuzzy subsets of vector spaces.

Properties like convexity, nonnegativity, and boundedness are naturally defined in cornets.

The main theorem extends the classical Cancellation Principle 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 R{\aa}dstr\"om in 1952.