Representation of semigroups as systems of compact convex sets

Authors:
H. Ratschek and G. Schröder

Journal:
Proc. Amer. Math. Soc. **65** (1977), 24-28

MSC:
Primary 20M30

DOI:
https://doi.org/10.1090/S0002-9939-1977-0486260-7

MathSciNet review:
0486260

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Abstract: Under Minkowski addition and scalar multiplication the system of all compact convex subsets of 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 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 , 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0486260-7

Keywords:
Representation of semigroups,
systems of compact convex sets

Article copyright:
© Copyright 1977
American Mathematical Society