Representation Theorem for Finite Distributive Lattices

1. INDUCTION IN A FINITE LATTICE Let L be a 1-sorted structure and let A, B be subsets of L. Let us observe that A⊆ B if and only if: (Def. 1) For every element x of L such that x ∈ A holds x ∈ B. Let L be a lattice. Observe that there exists a chain of L which is non empty. Let L be a lattice and let x, y be elements of L. Let us assume that x ≤ y. A non empty chain of L is called a x-chain of y if: (Def. 2) x ∈ it and y ∈ it and for every element z of L such that z ∈ it holds x≤ z and z≤ y. We now state the proposition (1) For every lattice L and for all elements x, y of L such that x≤ y holds {x,y} is a x-chain of y.