Unions of admissible relations and congruence distributivity
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Abstract
We show that a variety V is congruence distributive if and only if there is some h such that the inclusion \alpha(R\circ R)\subseteq (\alpha R)^k holds in every algebra in V, where juxtaposition denotes intersection, varies among tolerances and varies among U-admissible relations, that is, binary relations which are set-theoretical unions of reflexive and admissible relations. For any fixed h, a Maltsev-type characterization is given for the above inclusion. The results suggest that it might be interesting to study the structure of the set of U-admissible relations on an algebra, as well as identities dealing with such relations.
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How to Cite
Lipparini, P.
(2018).
Unions of admissible relations and congruence distributivity.
Acta Mathematica Universitatis Comenianae, 87(2), 251-266.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/681/625
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References
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Universalis 9 (1979), 142|145.
[2] G. Czedli, E. K. Horvath, Congruence distributivity and modularity permit
tolerances, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 41 (2002),
39-42.
[3] B. Jonsson, Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110-121.
[4] B. Jonsson, Congruence varieties, Algebra Universalis 10 (1980), 355-394.
[5] A. Kazda, M. Kozik, R. McKenzie, M. Moore, Absorption and directed Jonsson terms, ArXiv:1502.01072, 1-17 (2015).
[6] P. Lipparini, On the Jonsson distributivity spectrum, arXiv:1702.05353 (2017), 1-17.
[7] P. Lipparini, On unions of admissible relations, arXiv:1704.02476v2, 1-11
(2017).
[8] P. Lipparini, Relation identities equivalent to congruence modularity,
arXiv:1704.05274, 1-9 (2017).
[9] S. T. Tschantz, More conditions equivalent to congruence modularity in Universal algebra and lattice theory (Charleston, S.C., 1984), 270-282, Lecture Notes in Math. 1149, Springer, Berlin, 1985.
[10] L. Zadori, Monotone Jonsson operations and near unanimity functions, Algebra Universalis 33 (1995), 216-236.