Recognizable sets and power sets of finite semigroups

Let S be a finite semigroup or monoid. Then P(S), the power set of S, forms a finite semigroup or monoid under the usual multiplication of subsets. Let V be a variet~ of finite semigroups or monoids (~.~., ~ is a collection of finite semigroups, or monoids, closed under division and finite direct products) and let P(V) be the variety generated by (P(S)IS ~). In this paper I show that the operation ~ ~ P(V) on varieties is equivalent to a simple operation on the corresponding families of recognizable sets. Section ~ presents the necessary background material on varieties of recognizable sets and of finite semigroups and monoids--this material is extracted from Eilenberg [7, vol. B]. Section 2 contains a proof of the main theorem, which describes the effect of the operation ~ ~ P(V) on the variety of recognizable sets corresponding to ~. Section 3 is devoted to applications and some open questions.