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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 for all a, b E A. This theorem is a contribution to the concrete representation problem for congruence lattices. Other results related to this problem can be found in [2]. We always assume that a complete sublattice of a complete lattice U has the same extremal elements as U. Suppose t# is an equivalence relation on the set A and g is a function from some subset X of A3 into A. We write (a, b, c) i (d, e, ]) for a t# d, b t e, c t# f; g is compatible with t# provided g(a, b, c) i g(d, e, f) whenever (a, b, c) E X, (d, e, f) E X and (a, b, c) t (d, e, f); g is compatible with a set L of congruence relations if it is compatible with each member of L. Analogous definitions are used for unary functions. In the proof of Theorem 1 we use: THEOREM 2. Let L be a comwlete distributive lattice of permutable equivalence relations on the countable set A. There is a function f from A 3 into A which is compatible with L and for which (1) holds for all a, b E A. An analogous theorem, for A arbitrary and L finite, was proved in [3] (L was considered as a lattice of congruences of some algebra; however this fact was not substantially used). At the Colloquium on Universal Algebra in Oberwolfach, July 1973, A. F. Pixley asked whether the finiteness of L can be omitted in his theorem. Our theorems give a partial answer to his question. PROOF OF THEOREM 2. For the sake of convenience we suppose that A is countably infinite. The finite case may be obtained by halting our construction of f below at the appropriate place, but this case was fully established in [3] (cf. Lemma 3.1 where the assumption that L is the Received by the editors January 28, 1976 and, in revised form, October 22, 1976. AMS (MOS) subject classifications (1970). Primary 08A25. (? Amerncan Mathematical Society 1978

A ternary function for distributivity and permutability of an equivalence lattice