  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.
