Address: office 1410a, 17 Naberezhnaya Severnoy Dviny, Arkhangelsk, 163002, Russian Federation, Northern (Arctic) Federal University named after M.V. Lomonosov

Phone: (818-2) 21-61-21
E-mail: vestnik_est@narfu.ru
http://aer.narfu.ru/en/

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

Section: Physics. Mathematics. Informatics

512.541

### Authors

Institute of Information Technology, Cherepovets State University (Cherepovets, Russia)
е-mail: mukhinv1945@yandex.ru; mukhin@chsu.ru
Vologda Institute of Law and Economics of the Federal Penitentiary Service of Russia (Vologda, Russia)
е-mail: dina_sergeeva@mail.ru

### Abstract

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.

### Keywords

character, n-ary semigroup, topology, invariant measure