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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
MathSciNet review: 0486260
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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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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0486260-7
PII: S 0002-9939(1977)0486260-7
Keywords: Representation of semigroups, systems of compact convex sets
Article copyright: © Copyright 1977 American Mathematical Society