Rationality in Algebras with a Series Operation

Abstract

We consider the notion of rationality in algebras with a designated binary associative operation called the series operation, or the sequential product. We define automata operating in these algebras and rational expressions matching their expressive power, and we show that this expressive power equals that of algebraic recognizability. The framework which we consider encompasses both the free semigroup and the term algebras and the restriction of our results to these special cases coincides exactly with the classical results on recognizability (Kleene, Myhill, and Nerode for word languages, and Thatcher and Wright for term languages). Next we consider the behavior of the automata and the rational expression which we introduce when conditions such as associativity and commutativity are imposed on the term operations. We also characterize algebraically, syntactically and automata-theoretically the languages which have a bound on the number of nested occurrences of certain designated term operations. Finally, we consider the applications of our results to the languages of series-parallel labelled posets.