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A note on representation of lattices by tolerances. (English) Zbl 0777.08002

A tolerance on an algebra \(A\) is a reflexive and symmetric subalgebra of the Cartesian square of \(A\). Theorem: Let \(L\) be an algebraic lattice. Then there is an algebra \(A\) such that \(L\) is isomorphic to the lattice of all tolerances on \(A\); moreover, every subalgebra of \(A^ 2\) is a tolerance.

MSC:

08A30 Subalgebras, congruence relations
06B15 Representation theory of lattices
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References:

[1] Chajda, I., Recent results and trends in tolerances on algebras and varieties, (Finite Algebra and Multiple-valued Logic, Szeged 1979. Finite Algebra and Multiple-valued Logic, Szeged 1979, Colloq. Math. Soc. János Bolyai, Vol. 28 (1981), North-Holland: North-Holland Amsterdam), 69-95 · Zbl 0484.08002
[2] Grätzer, G.; Lampe, W. A., On subalgebra lattices of universal algebras, J. Algebra, 7, 263-270 (1967) · Zbl 0178.01101
[3] Iskander, A. A., The lattice of correspondences of universal algebras, Izv. Akad. Nauk SSSR Ser. Mat., 29, 1357-1372 (1965), [Russian]
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