Power semigroups: results and problems
João Pacheco de Almeida · 1999
This is a survey of work on the algebraic theory of the power operator on pseudovarieties of semigroups and monoids. Besides exploring connections with other operators, various problems and partial solutions are presented. Recent generalizations of the power operator, particularly the one resulting from looking only at group-pointlike subsets of finite monoids, are also considered. 1. Introduction For any algebraic structure S, one may define on the power set P(S) a similar algebraic structure by letting the result of applying an operation to a bunch of subsets of S be the set of all operation values which can be obtained by replacing each set by one of its elements. The question that is then naturally raised concerns which properties are preserved in this construction. In the context of Universal Algebra, this question has been considered by looking at equational properties or, equivalently, by defining the power operator on varieties and looking for fixed points. The answer turns ou...