A ternary function for distributivity and permutability of an equivalence lattice

Abstract:

The main result of the paper is Theorem 1. Let A be a countable set and L be a complete sublattice of the equivalence lattice on A. The following are equivalent (i) L is a distributive lattice of permutable equivalence relations. (ii) There is an algebra with congruence lattice L among the fundamental operations of which is a ternary function f with the property \begin{equation}\tag {$1$} \quad f(a,b,b) = f(a,b,a) = f(b,b,a) = a\end{equation} for all $a, b \in A$.