The complete principal divisor lattices

In this paper, we introduce a concept of principal divisor lattice and describe the structure of its elements. We first give a necessary and sufficient condition for the existence of irredundant join irreducible decompositions in complete principal divisor distributive lattices, and prove that the complete lower continuous, principal divisor lattices have irredundant join irreducible decompositions. In the end, we show the descriptions of lattices that have unique (resp. replaceable) irredundant join irreducible decompositions in complete lower continuous principal divisor lattices.