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Commutative semigroups with cancellation law: a representation theorem. (English) Zbl 1251.22003

Minkowski-Rådström-Hörmander spaces are useful in mathematics and more specially in optimization and the differentiating of multifunctions (see [V. F. Dem’yanov and A. M. Rubinov, Quasidifferential calculus. Translations Series in Mathematics and Engineering. New York: Optimization Software, Inc., Publications Div. xi, 289 p. (1986; Zbl 0712.49012); Quasidifferentiability and related topics. Nonconvex Optimization and Its Applications. 43. Dordrecht: Kluwer Academic Publishers. xix, 391 p. (2000; Zbl 0949.00047)]. By this motivation, the authors study these spaces within a general framework.
Note that in this paper, by “a semigroup with 0”, the authors mean a semigroup with an identity 0 and not the zero element. Let \((S,+)\) be a commutative semigroup with neutral element 0 satisfying the cancellation law and let \(S_0\) denote the group of units of the semigroup \(S\). A function \(T:S\to S\) is called a semigroup symmetry, if \(T\) is additive, \(T^2=Id_S\) and \(T|_{S_0}=-Id_{S_0}\), where \(S_0\) denotes the group of units of the semigroup \(S\). The semigroup \((S,+)\) is called 2-torsion-free if \(s+s=t+t\) implies \(s=t\). Denoting \(Ts\) by \(s^T\), a symmetry \(T\) is 2-divisible if for all \(s\in S\) we have \(s+s^T=t+t\) for some \(t\in S\). For \((s,t),(s',t')\in S\times S\) let \((s,t)\sim (s',t')\) if and only if \(s+t'= s'+t\). Denote \(\tilde{S}=S^2/\sim\). It is proved that \(\tilde{S}\) by natural addition on equivalence classes is a group and \(S\) can be embedded into \(\tilde{S}\) via \(s\mapsto \tilde{s}=[s,0]\), where \([s,0]\) denotes the equivalence class which contains \((s,0)\).
As the main results of the paper, after defining the notion of semigroup symmetry \(T\), the authors define a symmetric subgroup \(\tilde{S}_s\) and an asymmetric subgroup \(\tilde{S}_a\) of the topological group \(\tilde{S}\). Then they decompose the topological group \(\tilde{S}\) into a topological direct sum of its symmetric subgroup \(\tilde{S}_s\) and its asymmetric subgroup \(\tilde{S}_a\). Furthermore, they prove that for a 2-torsion-free semigroup \(S\) and a 2-divisible symmetry \(T\), the decomposition of elements of \(\tilde{S}\) into a sum of elements of the symmetric subgroup \(\tilde{S}_a\) and the asymmetric subgroup \(\tilde{S}_a\) is polar. Finally, they give conditions under which a topological group \(\tilde{S}\) is a topological direct sum of its symmetric subgroup \(\tilde{S}_s\) and its asymmetric subgroup \(\tilde{S}_a\).

MSC:

22A15 Structure of topological semigroups
20M14 Commutative semigroups
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