The arithmetic of a semigroup of series of Walsh functions

Let W = {w(k)(t)}(k=0)(infinity), be the classical system of the Walsh func tions, L-W the multiplicative semigroup of the functions represented by ser ies of functions wk (t) with non-negative coefficients which sum equals 1. We study the arithmetic of L-W. The analogues of the well-known Khinchin fa ctorization theorems related to the arithmetic of the convolution semigroup of probability measures on the real line are valid in L-W. The classes of idempotent elements, of infinitely divisible elements, of elements without indecomposable factors, and of elements without indecomposable and non-dege nerate idempotent factors are completely described. We study also the class of indecomposable elements. Our method is based on the following fact: L-W is isomorphic to the semigroup of probability measures on the group of cha racters of the Cantor-Walsh group. 2000 Mathematics subject classification: primary 60B15, 43A25; secondary 42C10.