Measure Algebras and Functions of Bounded Variation on Idempotent Semigroups

Abstract

Our main result establishes an isomorphism between all functions on an idempotent semigroup $S$ with identity, under the usual addition and multiplication, and all finitely additive measures on a certain Boolean algebra of subsets of $S$, under the usual addition and a convolution type multiplication. Notions of a function of bounded variation on $S$ and its variation norm are defined in such a way that the above isomorphism, restricted to the functions of bounded variation, is an isometry onto the set of all bounded measures. Our notion of a function of bounded variation is equivalent to the classical notion in case $S$ is the unit interval and the “product” of two numbers in $S$ is their maximum.