A ternary operation in distributive lattices
Garrett Birkhoff, S. A. Kiss · Bulletin of the American Mathematical Society · 1947
It can be easily seen that the graph [l, p. 9], 1 of the Boolean algebra B n of 2 n elements (consisting of the vertices and edges of an n-cube) has 2 n (n\) "link-automorphisms," whereas B n has only (n\) lattice-automorphisms.In an unpublished book, 2 one of us has developed new operations in B n and other distributive lattices, which admit such a wider group of invariance.The purpose of this note is to show the role of the symmetric and self-dual ternary operation [l.p.74] (a, b, c) « (aC\b)KJ (br\c)yj(cr\ a) = (aUJ)n(JUc)n.(cUa)in a general distributive lattice L, with reference to the wider group of symmetries which it admits.THEOREM 1.In any metric distributive lattice [l, p. 41], the following conditions are equivalent: (i) ar\b^x^a\Jb, (ii) \a-x\ + \x~b\ = | a -b\, (iii) (a, x, b) =#.PROOF.V. Glivenko [3, p. 819, Theorem V] has shown the equivalence of (i) and (ii) ; condition (i) says that x is metrically "between" a and b in the sense of Menger.But now if aC\b^x^a\Jb, then (a, x, b) = (anb)\J(br\x)U(xna)~(ar\b)\J[xr\(aKJb)]=x.Conversely, if (a, x, b) -x, then * « (a H 6) U (ft H *) VJ (* H a) £ a H ft, and dually, x^aKJb.Hence (i) and (iii) are equivalent.DEFINITION.The segment joining a and b is the set of x satisfying any (hence all) of the conditions of Theorem 1 (cf.Duthie [4]); we denote it [a, 6].THEOREM 2. The element (a, 6, c) is the intersection of the sets [a, 6] [6, c] [c, a].PROOF.This is obvious from condition (i) and formula (1).COROLLARY 1.The element (a, 6, c) minimizes