The arithmetic of a semigroup of series of Walsh functions

Abstract

Let "W = {u> t (0)£i 0 t * t n e classical system of the Walsh functions, «5V the multiplicative semigroup of the functions represented by series of functions w^ (t) with non-negative coefficients which sum equals 1. We study the arithmetic of 5f-#/. The analogues of the well-known Khinchin factorization theorems related to the arithmetic of the convolution semigroup of probability measures on the real line are valid in 5^iti-The classes of idempotent elements, of infinitely divisible elements, of elements without indecomposable factors, and of elements without indecomposable and non-degenerate idempotent factors are completely described. We study also the class of indecomposable elements. Our method is based on the following fact: S^tf is isomorphic to the semigroup of probability measures on the group of characters of the Cantor-Walsh group.2000 Mathematics subject classification: primary 60B15,43A25; secondary 42C10.