Skip to main content

Axiomatizability of positive algebras of binary relations

Abstract

We consider all positive fragments of Tarski’s representable relation algebras and determine whether the equational and quasiequational theories of these fragments are finitely axiomatizable in first-order logic. We also look at extending the signature with reflexive, transitive closure and the residuals of composition.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Andréka, H.: On the representation problem of distributive semilattice-ordered semigroups. Tech. rep., Mathematical Institute of the Hungarian Academy of Sciences (1988). Abstracted in Abstracts of the American Mathematical Society, 10(2):174, March 1989

  2. 2.

    Andréka H.: Representation of distributive lattice-ordered semigroups with binary relations. Algebra Universalis 28, 12–25 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Andréka H., Bredikhin D.: The equational theory of union-free algebras of relations. Algebra Universalis 33, 516–532 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Andréka H., Mikulás Sz.: Lambek calculus and its relational semantics: completeness and incompleteness. Journal of Logic, Language and Information 3, 1–37 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Andréka H., Németi I.: Axiomatization of identity-free equations valid in relation algebras. Algebra Universalis 35, 256–264 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Andréka H., Németi I., Sain I.: Algebraic logic. In: Gabbay, D., Guenthner, F (eds) Handbook of Philosophical Logic, vol. 2, second edn., pp. 133–247. Kluwer Academic Publishers, Dordrecht (2001)

    Google Scholar 

  7. 7.

    Bredikhin, D.: The equational theory of relation algebras with positive operations. Izv. Vyash. Uchebn. Zaved. Math. pp. 23–30 (1993) (Russian)

  8. 8.

    Bredikhin D., Schein B.: Representation of ordered semigroups and lattices by binary relations. Colloquium Mathematicum 39, 1–12 (1978)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Conway J.: Regular Algebra and Finite Machines. Chapman and Hall, London (1971)

    MATH  Google Scholar 

  10. 10.

    Crvenković S., Dolinka I., Ésik Z.: The variety of Kleene algebras with conversion is not finitely based. Theoretical Computer Science 230, 235–245 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Ésik Z., Bernátsky L.: Equational properties of Kleene algebras of relations with conversion. Theoretical Computer Science 137, 237–251 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Haiman M.: Arguesian lattices which are not type 1. Algebra Universalis 28, 128–137 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Hirsch R.: The class of representable ordered monoids has a recursively enumerable, universal axiomatisation but it is not finitely axiomatisable. Logic Journal of the IGPL 13, 159–171 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Hirsch R., Hodkinson I.: Relation Algebras by Games. Elsevier, Amsterdam (2002)

    MATH  Google Scholar 

  15. 15.

    Hirsch R., Mikulás Sz.: Representable semilattice-ordered monoids. Algebra Universalis 57, 333–370 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Hirsch R., Mikulás Sz.: Positive reducts of relevance logic and algebras of binary relations. Review of Symbolic Logic 4, 81–105 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Hodkinson I., Mikulás Sz.: Axiomatizability of reducts of algebras of relations. Algebra Universalis 43, 127–156 (2000)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Jipsen P., Jónsson B., Rafter J.: Adjoining units to residuated Boolean algebras. Algebra Universalis 34, 118–127 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Kozen D.: A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110, 366–390 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Maddux R.: Relation Algebras. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  21. 21.

    Mikulás Sz.: Axiomatizability of algebras of binary relations. In: Löwe, B., Piwinger, B., Räsch, T (eds) Classical and New Paradigms of Computation and their Complexity Hierarchies., pp. 187–205. Kluwer Academic Publishers, Dordrecht (2004)

    Chapter  Google Scholar 

  22. 22.

    Monk J.: On representable relation algebras. Michigan Mathematics Journal 11, 207–210 (1964)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Németi I.: Every free algebra in the variety generated by the representable dynamic algebras is separable and representable. Theoretical Computer Science 17, 343–347 (1982)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Németi I.: Algebraizations of quantifier logics, an introductory overview. Studia Logica 50(3–4), 485–569 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Pratt V.: Action logic and pure induction. In: Eijck, J. (eds) Logics in AI: European Workshop JELIA ’90., pp. 97–120. Springer, Heidelberg (1990)

    Google Scholar 

  26. 26.

    Redko V.: On defining relations for the algebra of regular events. Ukrain. Mat. Z. 16, 120–126 (1964) (Russian)

    MathSciNet  Google Scholar 

  27. 27.

    Schein B.: Representation of subreducts of Tarski relation algebras. In: Andréka, H., Monk, J., Németi, I. (eds) Algebraic Logic., pp. 621–635. North-Holland, Amsterdam (1991)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Szabolcs Mikulás.

Additional information

Andréka’s research was supported by OTKA grant Nos. 73601 and 81188.

Presented by J. Raftery.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Andréka, H., Mikulás, S. Axiomatizability of positive algebras of binary relations. Algebra Univers. 66, 7 (2011). https://doi.org/10.1007/s00012-011-0142-3

Download citation

2010 Mathematics Subject Classification

  • Primary: 03G15
  • Secondary: 06F05

Key words and phrases

  • representable relation algebras
  • finite axiomatizability
  • ordered semigroups