Arithmetic of semigroups of series in multiplicative systems

Abstract

Il'inskaya UDC 519.21 We study the arithmetic of a semigroup M P of functions with operation of multiplication representable in the form f x a x n n n ( ) ( ) = = ∞ ∑ χ 0 a a n n n ≥ = = ∞ ∑ ( ) 0 1 0 , , 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.