The Disjunctivities of w-languages

An ω-language over a finite alphabet X is a set of infinite sequences of letters of X. Consider congruences IL and Pω, L on X* and a congruence OL on X introduced by an ω-language L. IL, Pω, L, and OL are called the infinitary syntactic-congruence, the principal congruence and the ω-syntactic congruence of L, respectively. If IL (Pω, L, OL) is the equality then L is called an I-disjunctive (P-disjunctive, O-disjunctive, respectively) ωlanguage. Properties concerning such ω-languages are explored and relations between these ω-languages are also studied. The disjunctivity concerning the infinitary syntactic-congruence IL Given an ω-language L, by the infinitary syntactic-congruence IL of L we mean the relation IL on X* given by u ≡ v (IL) ⇔ ∀ x, y ∈ X*, x(uy) ∈ L iff x(vy) ∈ L. If IL is the equality then L is called I-disjunctive. Every I-discrete I-dense ω-language is I-disjunctive. An ω-language is I-dense iff it contains an I-disjunctive language. A periodically generated ω-language L is I-dense iff L can be expressed as a disjoint union of infinitely many I-disjunctive ω-languages. The disjunctivity concerning the principal congruence Pω, L Given an ω-language L, by the principal congruence Pω, L of L we mean the relation Pω, L on X* given by u ≡ v (Pω, L) ⇔ ∀ x∈ X* and α ∈ X, xuα ∈ L iff xvα ∈ L. If Pω, L is the equality then L is called Pdisjunctive. Every P-discrete P-dense ω-language is P-disjunctive. A P-discrete ω-language is P-disjunctive iff the set of all its finite subwords is X*. The disjunctivity concerning the ω-syntactic congruence OL Given an ω-language L, by the ω-syntactic congruence OL of L we mean the relation OL on X given by α ≡ β (O L) ⇔ ∀ x∈ X*, xα ∈ L iff xβ ∈ L. If OL is the equality then L is called O-disjunctive. If S is a left singular language then SL is O-disjunctive for any O-disjunctive ω-language L. If P is a finite prefix code then L is an O-disjunctive ω-language iff PL is O-disjunctive. Families of Disjunctive ω-languagesEvery I-closed I-disjunctive ω-language is P-disjunctive while not every P-disjunctive ω-language is I-disjunctive. Every O-disjunctive ω-language is P-disjunctive while not every P-disjunctive ω-language is O-disjunctive. Every O-disjunctive ω-language is I-disjunctive while not every I-disjunctive ω-language is O-disjunctive. Every P-disjunctive ω-language is P-dense ReferencesC. Calude and S. S. Yu, Language-theoretic complexity of disjunctive sequences, Discrete Applied Math., 80(1997), 203-209.S. W. Jiang, H. J. Shyr and S. S. Yu, Completely disjunctive languages, Periodica Polytechnica TransportationEngineering, Proceedings of the 2n International Mathematical Mini Conference, Budapest 1988, Part II, Vol19(1991) 1-2, 101-110. H. Jurgensen, H. J. Shyr and G. Thierrin, Disjunctive ω-languages, J. of Information Processing andCybernetics EIK 19 (1983) 6, 267-278. 144R.L.I.M.S. Volume 1. Sept. 2000 H. Jurgensen and G. Thierrin, Which monoids are syntactic monoids of ω-languages?, J. of InformationProcessing and Cybernetics EIK 22 (1986), 513-526.H. Jurgensen and G. Thierrin, Some structural properties of ω-languages, in 13 Nat. School with Interat.Participation “Applications of mathematics in Technology”, Sofia (1988), 56-63.Lila Kari and G. Thierrin, Omega-syntactic congruences, J. of Automata, Languages and Combinatorics, 1(1996) 1, 13-26.R. C. Lyndon and m. P. Schutzenberger, The equation a = bc in a free group, Michigan Math. J., 9, (1962),289-298. Oded Maler and Ludwig Staiger, On syntactic congruences for ω-languages, Theoret. Comput. Sci., 183(1997), 93-112.H. J. Shyr, Free monoids and languages, 2 edt, Hon Min Book Company, Taichung, Taiwan, 1991. H. J. Shyr, A characterzation of dense languages, Semigroup Forum, vol. 30 (1984), 237-240.H. J. Shyr and D. C. Tseng, Some properties of dense languages, Soochow Journal of Mathematics, 10 (1984),127-131. L.Staiger, research in the theory of ω-languages, J. Comput. System Sci., 27 (1983), 434-448.