Generalized Rational Relations and their Logical Definability

The family of rational subsets of a direct product of free monoids Σ∗ 1×. . .×Σ ∗ n (the rational relations) is not closed under Boolean operations, except when n = 1 or when all Σi’s are empty or singletons. In this paper we introduce the family of generalized rational subsets of an arbitrary monoid as the closure of the singletons under the Boolean operations, concatenation and Kleene star (just adding complementation to the usual rational operations). We show that the monadic second order logic enriched with a predicate comparing the cardinalities can express all generalized rational relations. The converse, to wit all subsets defined by this logic are generalized rational subsets, is an open question.