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On difunctionality of class relations


For a given variety \({\mathcal {V}}\) of algebras, we define a class relation to be a binary relation \(R\subseteq S^2\) which is of the form \(R=S^2\cap K\) for some congruence class K on \(A^2\), where A is an algebra in \( {\mathcal {V}}\) such that \(S\subseteq A\). In this paper we study the following property of \({\mathcal {V}}\): every reflexive class relation is an equivalence relation. In particular, we obtain equivalent characterizations of this property analogous to well-known equivalent characterizations of congruence-permutable varieties. This property determines a Mal’tsev condition on the variety and in a suitable sense, it is a join of Chajda’s egg-box property as well as Duda’s direct decomposability of congruence classes.

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  1. Agliano, P., Ursini, A.: Ideals and other generalizations of congruence classes. J. Aust. Math. Soc. 53, 103–115 (1992)

    Article  MathSciNet  Google Scholar 

  2. Carboni, A., Pedicchio, M.C., Pirovano, N.: Internal graphs and internal groupoids in Mal’tsev categories. Can. Math. Soc. Conf. Proc. 13, 97–109 (1992)

    MATH  Google Scholar 

  3. Chajda, I.: The egg-box property of congruences. Math. Slovaca 38, 243–247 (1988)

    MathSciNet  MATH  Google Scholar 

  4. Chajda, I., Czédli, G., Horvaáth, E.K.: Trapezoid lemma and congruence distributivity. Math. Slovaca 53, 247–253 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Chajda, I., Eigenthaler, G., Länger, H.: Congruence Classes in Universal Algebra. Research Expositions in Mathematics, vol. 26. Heldermann Verlag, Berlin (2003)

    MATH  Google Scholar 

  6. Chajda, I., Halas, R.: Varieties satisfying the triangular scheme need not be conguence distributive. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 46, 19–24 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Day, A.: A characterization of modularity for congruence lattices of algebras. Can. Math. Bull. 12, 167–173 (1969)

    Article  MathSciNet  Google Scholar 

  8. Duda, J.: Varieties having directly decomposable congruence classes. Časopis pro pěstování matematiky 111, 394–403 (1986)

    MathSciNet  MATH  Google Scholar 

  9. Duda, J.: The upright principle for congruence distributive varieties. Abstract of a lecture seminar presented in Brno. (2000)

  10. Duda, J.: The triangular principle for congruence distributive varieties. Abstract of a lecture seminar presented in Brno. (2000)

  11. Fraser, G.A., Horn, A.: Congruence relations in direct products. Proc. Am. Math. Soc. 26, 390–394 (1970)

    Article  MathSciNet  Google Scholar 

  12. Gumm, H.P.: Congruence modularity is permutability composed with distributivity. Arch. Math. (Basel) 36, 569–576 (1981)

    Article  MathSciNet  Google Scholar 

  13. Gumm, H.P.: Geometrical methods in congruence modular algebras. Mem. Am. Math. Soc. (1983).

  14. Gumm, H.P., Ursini, A.: Ideals in universal algebra. Algebra Univ. 19, 45–54 (1984)

    Article  MathSciNet  Google Scholar 

  15. Jónsson, B., Tarski, A.: Direct Decompositions of Finite Algebraic Systems. Notre Dame Mathematical Lectures, vol. 5. University of Notre Dame, Indiana (1947)

    MATH  Google Scholar 

  16. Lambek, J.: Goursat’s theorem and the Zassenhaus lemma. Can. J. Math. 10, 45–56 (1957)

    Article  MathSciNet  Google Scholar 

  17. Mal’tsev, A.I.: On the general theory of algebraic systems. Mat. Sbornik N. S. 35, 3–20 (1954). (Russian)

    MathSciNet  MATH  Google Scholar 

  18. Pixley, A.F.: Distributivity and permutability in equational classes. Proc. Am. Math. Soc. 14, 105–109 (1963)

    Article  Google Scholar 

  19. Riguet, J.: Relations binaires, fermetures, correspondances de Galois. Bull. Soc. Math. Fr. 76, 114–155 (1948)

    Article  MathSciNet  Google Scholar 

  20. Ursini, A.: On subtractive varieties. I. Algebra Univ. 31, 204–222 (1994)

    Article  MathSciNet  Google Scholar 

  21. Werner, H.: A Mal’cev condition for admissible relations. Algebra Univ. 3, 263 (1973)

    Article  Google Scholar 

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We would like to thank the anonymous referees as well as Heinz-Peter Gumm, for their useful remarks on earlier versions of this paper.

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Correspondence to Michael Hoefnagel.

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Presented by Emil W. Kiss.

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Second author’s research is partially supported by the South African National Research Foundation. The third author acknowledges partial financial assistance by the Centre for Mathematics of the University of Coimbra—UID/MAT/00324/2019, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.

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Hoefnagel, M., Janelidze, Z. & Rodelo, D. On difunctionality of class relations. Algebra Univers. 81, 19 (2020).

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