Arithmetic of semigroups of series in multiplicative systems

  • I. P. Il’inskaya


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.
How to Cite
Il’inskaya, I. P. “Arithmetic of Semigroups of Series in Multiplicative Systems”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, no. 7, July 2009, pp. 939–947,
Research articles