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
(
1
)
f
(
a
,
b
,
b
)
=
f
(
a
,
b
,
a
)
=
f
(
b
,
b
,
a
)
=
a
\begin{equation}\tag {$1$} \quad f(a,b,b) = f(a,b,a) = f(b,b,a) = a\end{equation}
for all
a
,
b
∈
A
a, b \in A
.