In this paper we show that the ternary operation of a metric ternary distributive semi-lattice, a generalization of the ternary Boolean algebra of Grau [2], uniquely minimizes ternary distance. This generalizes a result of Birkhoff and Kiss [l, Corollary 1, p. 749]. We show, conversely, that in a metric space unique minimizing of ternary distance determines a ternary operation with respect to which the space is a ternary distributive semi-lattice. Particularly, a lattice whose graph satisfies the unique minimal ternary distance condition and certain finiteness conditions must be distributive.This answers a question proposed by Birkhoff and Kiss [l, p. 750].1. Definitions and postulates. We state our results at the close of this section.A ternary distributive semi-lattice, hereinafter abbreviated TDSL, is a set of 3 elements closed with respect to a ternary operation (a, b, c) satisfying the following identities.(Tl) (a,a,b)=a.(T2) (a, b, c) is invariant under all 6 permutations.(T3) (a, (b, c, d), e) = ((a, b, e), c, (a, d, e)).

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Let P be the partially-ordered set of join-irreducibles of a finite lattice L and let tl ~ {xE P l x g_ aE L}. In an earlier study [1] We shall employ ) , 2, > for proper inclusion, inclusion, and covering respectively; u and z for unit and zero elements; and prime notation for indicating a dual statement or condition. Particularly, the condition