On a lattice with a valuation
Junji Hashimoto · Proceedings of the American Mathematical Society · 1952
A real-valued function v(x) defined on a lattice is called a valuation if and only if it satisfies (1) v(x) + v(y) = v(x C' y) + v(x U y), and a distributive valuation if and only if it satisfies (2,) 2{v(xU yJ Uz)-v(xr y rz)} = v(x J y) + v(y.J z) + v(z u x) v(xf l y) v(y z) v(z n x). If z=xY.y, then (2) becomes (1); therefore a distributive valuation is a valuation. In his book Lattice theory, G. Birkhoff conjectured the following theorem. THEOREM 1. If L is a lattice with a distributive valuation which is not constant on an interval [x, y] unless x=y, then L is distributive.' We intend to affirm this proposition by proving more precisely the following theorem. THEOREM 2. A lattice L is distributive if and only if the following condition (*) is satisfied: (*) For every x <y in L, there exists a distributive valuation which is defined on L and is not constant on the interval [x, y]. We now begin with a lemma. LEMMA 1. A lattice L is modular if, for every x <y in L, there exists a valuation which is defined on L and is not constant on the interval [x, y]. PROOF. If L is not modular, then L contains a nonmodular fiveelement sublattice, in which akJb = a Jc = e, a \b = aG\c=f, b < c. If t is an element of the interval [b, c], then every valuation v(t) satisfies v(t) + v(a) = v(t n a) + v(t U a) = v(f) + v(e); hence v(t) is constant on the interval [b, c].