Not congruence-permutable lattice

Question

What is an example of a lattice having congruences that don't permute.

Equivalently, a lattice $L$ such that there exist $\theta, \sigma \in Con L$ for which $\theta \vee \sigma = \theta \circ \sigma$ does not hold.

Many thanks.

Answer 1

Well, it turns out to be easy...

For example, let $L = \{ 0, a, b, 1 \}$ be a chain, with $0<a<b<1$. Let the non-trivial blocks of $\theta$ be $\{ 0, a \}$ and $\{ b, 1 \}$ and the only non-trivial block of $\sigma$ be $\{ a, b \}$.

Then $(0,1) \in \theta \vee \sigma \setminus \theta \circ \sigma$.