On a Certain Property of the Elements of a Finitely Generated Lattice

Abstract

Firstly, a reminder that a lattice is a set S of elements; it is partially ordered, closed in relation to two lattice operations: the reunion a + b and the intersection a • b of any two elements a and b from set S. (The reunion a + b is the smallest element of the lattice containing both elements a and b; the intersection a • b is the greatest element of the lattice contained in both elements a and b. Obviously, a ≤ a + b, b ≤ a + b, a ≥ a • b, b ≥ a • b). A lattice may also be defined thusly: the generating elements of the lattice are given. Other elements, different from the generators, are obtained via the two lattice operations, applied to the generators.