algebra universalis

, Volume 33, Issue 4, pp 516–532 | Cite as

The equational theory of union-free algebras of relations

  • H. Andréka
  • D. A. Bredikhin


We solve a problem of Jónsson [12] by showing that the class ℛ of (isomorphs of) algebras of binary relations, under the operations of relative product, conversion, and intersection, and with the identity element as a distinguished constant, is not axiomatizable by a set of equations. We also show that the set of equations valid in ℛ is decidable, and in fact the set of equations true in the class of all positive algebras of relations is decidable.


Binary Relation Identity Element Relative Product Equational Theory Distinguished Constant 
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  1. [1]
    Ackermann, W.,Solvable cases of the decision problem, Studies in logic and the foundations of mathematics, North-Holland Publishing Co., Amsterdam, 1954, viii+114 pp.Google Scholar
  2. [2]
    Andréka, H.,On union-relation composition reducts of relation algebras, Abstracts of Amer. Math. Soc.10,2 (1989), 174.Google Scholar
  3. [3]
    Andréka, H.,Representation of distributive-lattice-ordered semigroups with binary relations, Algebra Universalis28 (1991), 12–25.Google Scholar
  4. [4]
    Andréka, H. andNémeti, I.,Positive reducts of representable relation algebras do not form a variety, Preprint (1990).Google Scholar
  5. [5]
    Börner, F. andPöschel, R.,Clones of operations on binary relations, Contributions to general algebra7 (1991), 51–70.Google Scholar
  6. [6]
    Bredikhin, D. A.,On relation algebras with general superpositions, in: Algebraic Logic, Coll. Math. Soc. J. Bolyai Vol. 54, North-Holland, 1991, pp. 111–124.Google Scholar
  7. [7]
    Bredikhin, D. A.,The variety generated by ordered involuted semigroups of binary relations, Proc. Suslin Conf., Saratov, 1991, p. 27.Google Scholar
  8. [8]
    Bredikhin, D. A.,The equational theory of relation algebras with positive operations (In Russian.) Izv. Vyash, Uchebn. Zaved., Math., No 3, 1993, pp. 23–30.Google Scholar
  9. [9]
    Bredikhin, D. A. andSchein, B. M.,Representations of ordered semigroups and lattices by binary relations, Colloq. Math.39 (1978), 1–12.Google Scholar
  10. [10]
    Comer, S. D.,A remark on representable positive cylindric algebras, Algebra Universalis28 (1991), 150–151.Google Scholar
  11. [11]
    Haiman, M.,Arguesian lattices which are not linear, Bull. Amer. Math. Soc.16 (1987), 121–123.Google Scholar
  12. [12]
    Jónsson, B.,Representation of modular lattices and relation algebras, Trans. Amer. Math. Soc.92 (1959), 449–464.Google Scholar
  13. [13]
    Kozen, D.,On induction vs *-continuity, in:Logics of Programs, Lecture Notes in Computer Science 131, Springer Verlag, Berlin, 1982, pp. 167–176.Google Scholar
  14. [14]
    Németi, I.,Algebraizations of quantifier logics, An introductory overview, Version 10.2, Preprint, Mathematical Institute of the Hungarian Academy of Sciences, Budapest, 1992, Abstracted in: Studia Logica, vol L, No 3/4, 1991.Google Scholar
  15. [15]
    Schein, B. M.,Relation algebras and function semigroups, Semigroup Forum1 (1970), 1–62.Google Scholar
  16. [16]
    Schein, B. M.,Representation of involuted semigroups by binary relations, Fund. Math.82 (1974), 121–141.Google Scholar
  17. [17]
    Tarski, A. andGivant, S.,A formalization of set theory without variables, Colloquium Publications 41, American Mathematical Society, Providence, R.I., 1987, xxii +318 pp.Google Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • H. Andréka
    • 1
  • D. A. Bredikhin
    • 2
  1. 1.Mathematical InstituteBudapestHungary
  2. 2.SaratovRussia

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