## 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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