On syntactic congruences for ω—languages

For ω-languages several notions of syntactic congruence were defined. The present paper investigates relationships between the so-called simple (because it is a simple translation from the usual definition in the case of finitary languages) syntactic congruence and its infinitary refinements investigated by Arnold [Ar85]. We show that in both cases not every ω-language having a finite syntactic monoid is regular and we give a characterization of those ω-languages having finite syntactic monoids. As the main result we derive a condition which guarantees that the simple syntactic congruence and Arnold's syntactic congruence coincide and show that all ω-languages in the Borel class F_θ∩G_δ satisfy this condition. Finally we define an alternative canonical object for ω-languages, namely a family of right-congruence relations. Using this object we give a necessary and sufficient condition for a regular ω-language to be accepted by its minimalstate automaton.