Representation of semigroups as systems of compact convex sets

Ratschek, H.; Schröder, G.

doi:10.1090/S0002-9939-1977-0486260-7

Representation of semigroups as systems of compact convex sets
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by H. Ratschek and G. Schröder PDF

Proc. Amer. Math. Soc. 65 (1977), 24-28 Request permission

Abstract:

Under Minkowski addition and scalar multiplication the system of all compact convex subsets of ${R^n}$ is an R-semigroup, i.e. a semigroup over the operator domain R of real numbers with certain conditions for the operation of R on the semigroup. Conversely, there is the question: When is an abstract R-semigroup isomorphic to a system $\mathfrak {S}$ of compact convex subsets of a suitable locally convex space? In this paper a necessary and sufficient condition for the existence of such a representation is given. This condition remains valid if, for the representing structures $\mathfrak {S}$, systems of closed, bounded convex subsets with the closed Minkowski addition as addition are permitted. Finally, every R-semigroup of compact convex subsets of any locally convex space is isomorphic to a system of rectangular parallelepipeds of some vector space.

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