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On tolerance lattices of algebras in congruence modular varieties. (English) Zbl 1049.08007

We recall that a lattice \(L\) with \(0\) is called \(0\)-modular if there is no \(N_5\) sublattice of \(L\) including \(0\); a bounded lattice \(L\) is called \(0-1\) modular if no \(N_5\) sublattice of \(L\) includes both \(0\) and \(1\).
The purpose of this paper is to extend known results on tolerance lattices of lattices to tolerance lattices of more generalized algebras. The authors prove that the tolerance lattice \(\operatorname{Tol} A\) of an algebra \(A\) from a congruence modular variety \(V\) is \(0-1\) modular and satisfies the general disjointness property (that is, if \(\alpha, \beta, \gamma \in \operatorname{Tol} A\), \(\alpha\wedge\beta = (\alpha \vee \beta)\wedge \gamma = 0\) imply \(\alpha\wedge (\beta\vee \gamma) = 0\)).
If \(V\) is congruence distributive, then the lattice \(\operatorname{Tol} A\) is pseudocomplemented; if \(V\) admits a majority term, then \(\operatorname{Tol} A\) is \(0\)-modular.

MSC:

08B10 Congruence modularity, congruence distributivity
06B10 Lattice ideals, congruence relations
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