Commutative semigroups with cancellation law: a representation theorem |
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Authors: | J Grzybowski D Pallaschke H Przybycień R Urbański |
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Institution: | 1.Adam Mickiewicz University,Poznań,Poland;2.University of Karlsruhe,Karlsruhe,Germany |
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Abstract: | Any commutative, cancellative semigroup S with 0 equipped with a uniformity can be embedded in a topological group \(\widetilde{S}\). We introduce the notion of semigroup symmetry T which enables us to turn \(\widetilde{S}\) into an involutive group. In Theorem 2.8 we prove that if S is 2-torsion-free and T is 2-divisible then the decomposition of elements of \(\widetilde{S}\) into a sum of elements of the symmetric subgroup \(\widetilde{S}_{s}\) and the asymmetric subgroup \(\widetilde{S}_{a}\) is polar. In Theorem 3.7 we give conditions under which a topological group \(\widetilde{S}\) is a topological direct sum of its symmetric subgroup \(\widetilde{S}_{s}\) and its asymmetric subgroup \(\widetilde{S}_{a}\). Theorem 2.8 and Theorem 3.7 are designed to be useful tools in studying Minkowski–Rådström–Hörmander spaces (and related topological groups \(\widetilde{S}\)), which are natural extensions of semigroups of bounded closed convex subsets of real Hausdorff topological vector spaces. |
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