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Proceedings of the American Mathematical Society

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A ternary function for distributivity and permutability of an equivalence lattice


Author: Ivan Korec
Journal: Proc. Amer. Math. Soc. 69 (1978), 8-10
MSC: Primary 08A25; Secondary 06A35
DOI: https://doi.org/10.1090/S0002-9939-1978-0472648-8
MathSciNet review: 0472648
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Abstract: The main result of the paper is

Theorem 1. Let A be a countable set and L be a complete sublattice of the equivalence lattice on A. The following are equivalent

(i) L is a distributive lattice of permutable equivalence relations.

(ii) There is an algebra with congruence lattice L among the fundamental operations of which is a ternary function f with the property

$\displaystyle \quad f(a,b,b) = f(a,b,a) = f(b,b,a) = a$ ($ 1$)

for all $ a, b \in A$.

References [Enhancements On Off] (What's this?)

  • [1] G. Grätzer, Universal algebra, Van Nostrand, Princeton, N. J., 1968. MR 0248066 (40:1320)
  • [2] B. Jonsson, Topics in universal algebra, Lecture Notes in Math., vol. 250, Springer-Verlag, Berlin and New York, 1970. MR 0345895 (49:10625)
  • [3] A. F. Pixley, Completeness in arithmetical algebras, Algebra Universalis 2 (1972), 179-196. MR 0321843 (48:208)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0472648-8
Article copyright: © Copyright 1978 American Mathematical Society

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