Omega-Syntactic Congruences

An ω-language over a finite alphabet X is a set of infinite sequences of letters of X. Previously studied syntactic equivalence relations defined by ω-languages have mainly been relations on X∗. In this paper the emphasis is put on relations in Xω, by associating to an ω-language L a congruence on Xω, called the ω-syntactic congruence of L. Properties of this congruence and notions induced by it, such as ω-residue, ω-density, and separativeness are defined and investigated. Finally, a congruence on X ∗ related to the ω-syntactic congruence and quasi-orders on Xω induced by an ω-language are studied.