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Continuous Characters of Topological Abelian Cancellative n-аry Semigroups. P. 117–124

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Section: Physics. Mathematics. Informatics




Mukhin Vladimir Vasil’evich
Institute of Information Technology, Cherepovets State University (Cherepovets, Russia)
Sergeeva Dina Vladimirovna
Vologda Institute of Law and Economics of the Federal Penitentiary Service of Russia (Vologda, Russia)


The paper studies the homomorphisms of topological Abelian cancellative n-ary semigroups into the group under multiplication of all complex numbers of modulus 1. These mappings are called characters. The set of all continuous characters of topological n-ary semigroup X is denoted ˆX . The set ˆX is a binary group with respect to the pointwise multiplication of characters. A preliminary result shows that the Abelian cancellative n-ary semigroup X can be considered as the n-ary subsemigroup of the n-ary group G, which as well as in the binary variant can be called the n-ary group of quotients of Abelian cancellative n-ary semigroup. Theorem 1 demonstrates that every character of Abelian n-ary semigroup naturally extends to the character on the n-ary group of its quotients. The group ˆX is endowed with the topology of uniform convergence on compact sets. Theorem 2 establishes that this topology is correlated with the group structure; i.e. ˆX becomes a topological binary group. Theorem 3 demonstrates the conditions of algebraic and topological isomorphism of the group ˆX to the group ˆG . The group of continuous characters of the binary group ˆX is denoted by the symbol ˆˆX . By analogy with the binary variant we consider a natural mapping π from X into ˆˆX that the character  π(x) of the group ˆX relates for every x of X in accordance with the formula π(x)(χ) = χ(x) (χ∈ˆX). Theorem 4 establishes that if there is a non-zero invariant Borel measure on a topological Abelian cancellative n-ary semigroup X, so the mapping π is continuous and injective, X has such nonvacuous open set U that restriction π to U is a homeomorphism of U onto the open subset π(U) of the group ˆX.


character, n-ary semigroup, topology, invariant measure
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