A hierarchy theorem for regular languages over free bisemigroups

In this article a question left open in [2] is answered. In particular, we show that it is essential that in the definition of parenthesizing automata an arbitrary number of parentheses can be used. Moreover, we prove that the classes Regm of languages accepted by a parenthesizing automaton with at most m pairs of parentheses form a strict hierarchy. In fact, this hierarchy is proper for all alphabets

Automata on infinite biposets

Bisemigroups are algebras equipped with two independent associative operations. Labeled finite sp-biposets may serve as a possible representation of the elements of the free bisemigroups. For finite sp-biposets, an accepting device, called parenthesizing automaton, was introduced in [6], and it was proved that its expressive power is equivalent to both algebraic recognizability and monadic second order definability. In this paper, we show, how this concept of parenthesizing automaton can be generalized for infinite biposets in a way that the equivalence of regularity (defined by acceptance with automata), recognizability (defined by homomorphisms and finite ω-bisemigroups) and MSO-definability remains true