Arithmetic of semigroups of series in multiplicative systems

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.