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Axiomatizability of positive algebras of binary relations

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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.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  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 

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Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 13–15 Reáltanoda u., 1053, Budapest, Hungary

    Hajnal Andréka

  2. Department of Computer Science and Information Systems, Birkbeck, University of London, Malet Street, London, WC1E 7HX, UK

    Szabolcs Mikulás

Authors
  1. Hajnal Andréka
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  2. Szabolcs Mikulás
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Correspondence to Szabolcs Mikulás.

Additional information

Presented by J. Raftery.

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

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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

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