Skip to main content
SpringerLink
Log in
Book cover

International Conference on Algebraic Informatics

CAI 2007: Algebraic Informatics pp 260–274Cite as

Nonassociativity à la Kleene

  • Conference paper

  • 381 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4728))

Abstract

First we recall the work of Suschkewitsch (1929) about the generalization of the associative law which is the starting point of the theory of quasigroups. Then we show that it is a particular case of the notion of relative associativity introduced by Roubaud in 1965. Thereafter we prove a coherence theorem over an infinite set of nonassociative operations. This result contains all the uppermentioned contributions. This allows to obtain a very general à-la-Kleene theorem on rational series which uses concatenations that can be associative or not.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ackél, J.: Lectures on Functional Equations and their Applications. Academic Press, New-York (1966)

    Google Scholar 

  2. Ackél, J., Hosszú, M.: On transformations with several parameters and operations in multidimensional spaces. Acta Math. Acad. Sci. Hungaricae 7, 327–338 (1956)

    Article  Google Scholar 

  3. d’Adhémar, C.: Quelques classes de groupoídes non-associatifs. Math. Sci. Hum. 31, 17–31 (1970)

    Google Scholar 

  4. Alexandrakis, A., Bozapalidis, S.: Weighted grammars and Kleene’s theorem. Inform. Process. Lett. 24, 1–4 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Berstel, J., Reutenauer, C.: Rational Series and their Langages. In: EATCS Monographs on Theoretical Computer Science, vol. 12, Springer, Heidelberg (1988)

    Google Scholar 

  6. Bozapalidis, S.: Context-free series on trees. Inform. Comput. 169, 186–229 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gécseg, F., Steinby, M.: Tree Languages. In: Rosenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. Beyond Words, vol. 3, pp. 1–68. Springer, Heidelberg (1997)

    Google Scholar 

  8. Germain, C., Pallo, J.: Langages rationnels définis avec une concaténation non-associative. Theor. Comput. Sci. 233, 217–231 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Germain, C., Pallo, J.: Linear languages with a nonassociative concatenation. J. Autom. Lang. Comb. 7, 311–320 (2002)

    MATH  MathSciNet  Google Scholar 

  10. Harris, Z.: A Theory of Language and Information: a Mathematical Approach. Oxford University Press, Oxford (1991)

    Google Scholar 

  11. Hausmann, B.A., Ore, O.: Theory of quasigroups. Amer. J. Math. 59, 983–1004 (1937)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kostrikin, A.L., Shafarevich, I.R. (eds.): Algebra VI. Encyclopaedia of Mathematical Sciences, vol. 57. Springer, Heidelberg (1995)

    MATH  Google Scholar 

  13. Kuich, W.: Semirings and Formal Power Series: Their Relevance to Formal Languages and Automata. In: Rosenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. Word, Language, Grammar, vol. 1, pp. 609–677. Springer, Heidelberg (1997)

    Google Scholar 

  14. Kunen, K.: Quasigroups, loops and associative laws. J. Algebra 185, 194–204 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lõhmus, J.: Preface to the Special Issue on Nonassociative Algebras, Quasigroups and Applications in Physics. Acta Applic. Math. 50, 1–2 (1998)

    Article  Google Scholar 

  16. L\~{o}hmus, J., Paal, E., Sorgsepp, L.: About nonassociativity in mathematics and physics. Acta Applic. Math. 50, 3–31 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Mac Lane, S.: Natural associativity and commutativity. Rice Univ. Stud. 49, 28–46 (1963)

    MathSciNet  Google Scholar 

  18. Mac Lane, S.: Preface to ”Coherence in Categories”. Lecture Notes in Math., vol. 281. Springer, Heidelberg (1970)

    Google Scholar 

  19. Murdoch, D.C.: Quasigroups which satisfy certain generalized associative law. Amer. J. Math. 61, 509–522 (1939)

    Article  MATH  MathSciNet  Google Scholar 

  20. Pallo, J.M.: Word problem in distributive magmas. Fund. Inform. 4, 957–973 (1981)

    MATH  MathSciNet  Google Scholar 

  21. Pallo, J.M.: Modéles associatif-relatif et commutatif cohérents appliqués aux langages réguliers. Calcolo 19, 289–300 (1982)

    MATH  MathSciNet  Google Scholar 

  22. Pallo, J.M.: Coding binary trees by embedding into the Roubaud’s magma. Rad. Mat. 2, 21–34 (1986)

    MATH  MathSciNet  Google Scholar 

  23. Pallo, J.M.: The rotation χ-lattice of ternary trees. Computing 66, 297–308 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Pallo, J.M.: Generating binary trees by Glivenko classes on Tamari lattices. Inform. Process. Lett. 85, 235–238 (2003)

    Article  MathSciNet  Google Scholar 

  25. Pallo, J.M.: Permutoassociaédres d’arbres binaires étiquetés. Rad. Mat. 13, 5–14 (2004)

    MATH  MathSciNet  Google Scholar 

  26. Pflugfelder, H.O.: Historical notes on loop theory. Comment. Math. Univ. Carolinae 41, 359–370 (2000)

    MATH  MathSciNet  Google Scholar 

  27. Roubaud, J.: Types d’A-algèbres discrètes complètes: un théorème des fonctions implicites. C. R. Acad. Sc. Paris 261, 3005–3007 (1965)

    MathSciNet  Google Scholar 

  28. Roubaud, J.: Sur un théorème de M. P. Schützenberger. C. R. Acad. Sc. Paris 261, 3265–3267 (1965)

    MATH  MathSciNet  Google Scholar 

  29. Roubaud, J.: La notion d’associativité relative. Math. Sci. Hum. 34, 43–59 (1970)

    MathSciNet  Google Scholar 

  30. Sade, A.: Théorie des systèmes demosiens de groupoïdes. Pacific J. Math. 10, 625–660 (1960)

    MATH  MathSciNet  Google Scholar 

  31. Sade, A.: Groupoïdes en relation associative et semigroupes. Ann. Soc. Sc. Bruxelles 75, 52–57 (1961)

    MathSciNet  Google Scholar 

  32. Salomaa, A.: Formal Languages and Power Series. In: Van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 103–132. Elsevier, Amsterdam (1990)

    Google Scholar 

  33. Suschkewitsch, A.: On a generalization of the associative law. Trans. Amer. Math. Soc. 31, 204–214 (1929)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Bourgogne, LE2I UMR 5158, BP 47870, 21078 DIJON-Cedex, France

    Jean-Marcel Pallo

Authors
  1. Jean-Marcel Pallo
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Symeon Bozapalidis George Rahonis

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Pallo, JM. (2007). Nonassociativity à la Kleene. In: Bozapalidis, S., Rahonis, G. (eds) Algebraic Informatics. CAI 2007. Lecture Notes in Computer Science, vol 4728. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75414-5_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-75414-5_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-75413-8

  • Online ISBN: 978-3-540-75414-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation