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.
Highlights
The aim of this paper is to introduce the notion of cornets, which form a particular subclass of ordered semigroups 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 the theory of convex sets, a basic Cancellation Principle was discovered by Rådström [50] in 1952
Summary
In the theory of convex sets, a basic Cancellation Principle was discovered by Rådström [50] in 1952. A+ B ⊆C + B implies A ⊆ C provided that A, B, C are nonempty subsets of a normed space X , C is closed and convex and B is bounded This lemma turned out to be a basic tool in various fields and hundreds of papers have used it . It turns out that the natural setting of the Cancellation Principle is a commutative ordered semigroup which is equipped with a multiplication by natural numbers. These structures will be termed cornets in our paper.
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