The Rees-Suschkewitsch Structure Theorem for Compact Simple Semigroups

Abstract

Kn(w) Kn(an)(W + o(1)) n-~~~~~+ o(1). (14) n n Then relations (14), (10), and the continuity of 9Z(w) yield relation (4), which completes the proof. Given an additive function f(m), define its strongly additive contraction by f(m) = E f(p). p/m One can easily prove THEOREM B. Let f(m) be an additive arithmetic function and f(m) its strongly additive contraction, and define AZ, B, as in equations (3). Then, assuming relation (5), if one of f(m)-A f(m)-An Bn Bn, has a continuous distribution function F(w), the other also has F(w) as its distribution function. Combining Theorems A and B yields THEOREM C. Relations (1), (5), and (7) imply relation (4).

Idempotent measures on semigroups

“…In §B we describe the structure of compact kernels, giving results obtaing by Wallace [9] as a preliminary to describing the idempotent measures on them in §C. The relationships between invariant and idempotent measures are given in Theorems C4.1 and C5.1.…”

“…The results of this section were given in a slightly more general form (see (4) below) by Wallace [9], We give them here in order to establish our notation, and because Wallace gave no proofs.…”

Limits of Convolution Sequences of Measures on a Compact Topological Semigroup

“…When studying K> it is natural to look at it with the relative topology, if is a completely simple semigroup, that is, KkK = K for all k z K and K has a minimal left and a minimal right ideal [8]. Every compact completely simple semigroup can be represented as the product space T X X X Y of a compact topological group T and compact Hausdorff spaces X and Y where the multiplication is given by…”

“…with <p a continuous function on the product space X X Y into T [8]. We shall therefore identify K with such a space T X X X Y and the accompanying <p function.…”

Idempotent measures on a compact topological semigroup

“…From this point on let us take 5 a compact completely simple semigroup with representation TXXX Fand corresponding function Suppose ¡t is an idempotent measure on 5 with support 5. Let (6) 5«= {*| *(*) = *}, that is, Sx is the subset of points in 5 whose x coordinate x(s) in representation ( ) of the semigroup is the fixed point x in X. Then P(s, A)-ß(As-1) is an idempotent Markov transition measure (see [4]) for AE®(SX), sESx, that is The proof of the theorem is complete.…”