We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Commutative semigroups with cancellation law: a representation theorem.
- Authors
Grzybowski, J.; Pallaschke, D.; Przybycień, H.; Urbański, R.
- 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.
- Subjects
ABELIAN semigroups; GROUP theory; TOPOLOGICAL groups; MATHEMATICAL decomposition; VECTOR topology; MATHEMATICAL analysis
- Publication
Semigroup Forum, 2011, Vol 83, Issue 3, p447
- ISSN
0037-1912
- Publication type
Article
- DOI
10.1007/s00233-011-9327-5