Arithmetic of semigroups of series in multiplicative systems - Il’inskaya, I. P. (Ukrains’kyi Matematychnyi Zhurnal, 2009)
We study the arithmetic of a semigroup M_P of functions with operation of multiplication representable in the form f(x)=∑^∞_n=0 a_nχ_n(x)(a_n≥0, ∑^∞_n=0a_n =1 ), where {χ_n|}^∞_n=0 is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup 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(M_P) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in M_P in the topology of uniform convergence.