Distributivity and decomposability on the lattices satisfying the chain conditions Original Research Article

This short note will confirm the converse of Birkhoffʹs theorem and establish the following: Suppose L is a lattice satisfying the ascending chain condition (resp., the descending chain condition). Then L is distributive if and only if each element of L has a unique finite irredundant decomposition into irreducibles. This conclusion has improved some related results of Dilworth (1961). In addition, some interesting corrollaries are presented.