Notes on tolerance relations of lattices: A conjecture of R. N. McKenzie (Q757456)

If \({\mathcal V}\) and \({\mathcal W}\) are lattice varieties, then \({\mathcal V}\circ {\mathcal W}\) consists of all lattices L for which there is a congruence \(\theta\) on L such that all \(\theta\)-classes of L are in \({\mathcal V}\) and L/\(\theta\in {\mathcal W}\). In general, \(V\circ W\) is not a variety. R. N. McKenzie conjectured that a lattice K belongs to the variety generated by \({\mathcal V}\circ {\mathcal W}\) iff there is a tolerance T on K such that all T- classes of L are in \({\mathcal V}\) and L/T\(\in {\mathcal W}\). The aim of this paper is to disprove this conjecture. It presents a finite (18 element) lattice F of the variety generated by \({\mathcal M}_ 3\circ {\mathcal D}\) which fails the mentioned conjecture.