2018
Том 70
№ 11

# Arithmetic of semigroups of series in multiplicative systems

Il’inskaya I. P.

Abstract

We study the arithmetic of a semigroup $\mathcal{M}_P$ of functions with operation of multiplication representable in the form $f(x)=∑^{∞}_{n=0} a_nχ_n(x)\left(a_n≥0,\; ∑^{∞}_{n=0}a_n =1 \right)$, where $\{χ_n|\}^{∞}_{n=0}$ is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup $\mathcal{M}_P$ , analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in $R_n$ are true. We describe the class $I_0(\mathcal{M}_P)$ of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in $\mathcal{M}_P$ in the topology of uniform convergence.

English version (Springer): Ukrainian Mathematical Journal 61 (2009), no. 7, pp 1113-1122.

Citation Example: Il’inskaya I. P. Arithmetic of semigroups of series in multiplicative systems // Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 939–947.

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