Tolerances and Commutators on lattices

Abstract

The commutator has the following order theoretic properties: [α, β] ≦ α ∧ β, [α, β] = [β α],[α1∨ α2,β] = [α1, β] ∨ [α2, β] for congruences α, β ∈ ConAof an algebraAin a congruence modular variety generalising the original concept in group theory. A tolerance of a latticeLis a reflexive and symmetric sublattice ofL2. We show that to every commutator [ , ] of ConAcorresponds a ∧-subsemilattice of the lattice of tolerances of ConA. It can be shown thatAin a congruence modular variety is nilpotent if |conA| > 2 and ConAis simple.