Acta Informatica

, Volume 48, Issue 1, pp 1–18 | Cite as

Some properties of the disjunctive languages contained in Q

Original Article


The set of all primitive words Q over an alphabet X was first defined and studied by Shyr and Thierrin (Proceedings of the 1977 Inter. FCT-Conference, Poznan, Poland, Lecture Notes in Computer Science 56. pp. 171–176 (1977)). It showed that for the case |X| ≥ 2, the set along with \({Q^{(i)} = \{f^i\,|\,f \in Q\}, i\geq 2}\) are all disjunctive. Since then these disjunctive sets are often be quoted. Following Shyr and Thierrin showed that the half sets \({Q_{ev} = \{f \in Q\,|\,|f| = {\rm even}\}}\) and Q od = Q \ Q ev of Q are disjunctive, Chien proved that each of the set \({Q_{p,r}= \{u\in Q\,|\,|u|\equiv r\,(mod\,p) \},\,0\leq r < p}\) is disjunctive, where p is a prime number. In this paper, we generalize this property to that all the languages \({Q_{n,r}= \{u\in Q\,|\,|u|\equiv r\,(mod\,n) \},\, 0\leq r < n}\) are disjunctive languages, where n is any positive integer. We proved that for any n ≥ 1, k ≥ 2, (Q n,0) k are all regular languages. Some algebraic properties related to the family of languages {Q n,r | n ≥ 2, 0 ≤ r < n } are investigated.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chien T.Y.: Decompositions of a free monoid into disjunctive languages. Soochow J. Math. 5, 121–127 (1979)MATHMathSciNetGoogle Scholar
  2. 2.
    Choffrut C., Karhumäki J.: Combinatorics of Words. In: Rozenberg, G., Salomaa, S. (eds) Handbook of Formal languages, Vol. 1, Ch. 6, pp. 329–438. Springer-verlag, Berlin (1997)Google Scholar
  3. 3.
    Dömösi P., Horväth G., Vuillon L.: On Shyr-Yu theorem. Theor Comput Sci 410, 4874–4877 (2009)MATHCrossRefGoogle Scholar
  4. 4.
    Dömösi, P., Horväth, S., Ito, M., Käszonyi, L., Katsura, M.: Some combinatorial properties of words, and the Chomsky-hierarchy. In: Ito, M., Jürgensen, H., (eds.) Words, languages and combinatorics, II (Kyoto, 1992), pp. 105–123, World Sci. Publishing, River Edge, NJ (1994)Google Scholar
  5. 5.
    Hungerford T.W.: Algebra. Springer-Verlag, New York (1974)MATHGoogle Scholar
  6. 6.
    Levi F.: On semigroups. Bull Calcutta Math. Soc. 36, 141–146 (1944)MATHMathSciNetGoogle Scholar
  7. 7.
    Lin, K.-N.: On some classes of languages and codes. Ph.D. thesis, Tamkang College of Arts and Sci (1979)Google Scholar
  8. 8.
    Lyndon R.C., Schützenberger M.P.: The equation a M = b N c P in a free group. Michgan Math. J. 9, 289–298 (1962)MATHCrossRefGoogle Scholar
  9. 9.
    Păun G., Santean N., Thierrin G., Yu S.S.: On the robusteness of primitive words. Discrete Appl. Math. 117, 239–252 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Reis C.M., Shyr H.J.: Some properties of disjunctive languages on a free monoid. Inf. Control 37(3), 334–344 (1978)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Shyr H.J.: Disjunctive languages on a free monoid. Inf. Control 34, 123–129 (1977)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Shyr, H.J., Thierrin, G.: Disjunctive languages and codes, fundamentals of computation theory. In: Proceedings of the 1977 Inter. FCT-Conference, Poznan, Poland, Lect. Notes in Comput. Sci. 56, 171–176 (1977)Google Scholar
  13. 13.
    Shyr H.J.: Free Monoids and Languages. 3rd edn. Hon Min Book Company, Taichung, Taiwan (2001)Google Scholar
  14. 14.
    Shyr H.J., Yu S.S.: Languages defined by two functions. Soochow J. Math. 20(3), 279–296 (1994)MATHMathSciNetGoogle Scholar
  15. 15.
    Shyr H.J., Yu S.S.: Non-primitive words in the language p + q +. Soochow J. Math. 20, 535–546 (1994)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Center of General EducationAletheia University on Motou CampusMatou, TainanTaiwan
  2. 2.Department of Applied MathematicsChung-Yuan Christian UniversityChung-LiTaiwan

Personalised recommendations