Unique factorization in partially ordered sets

Abstract:

Call an ideal of a lattice L regular if it is maximal with respect to not containing some element of L. If the ideal M is maximal without $x \in L$ we say M is a value of x. A special ideal K is a regular ideal which is maximal without some element having only one value (namely K). Our main theorem is that the following three statements are equivalent: (1) The lattice of ideals of the lattice L is completely distributive. (2) Each regular ideal is special. (3) L is distributive, and each element $x \in L$ can be written uniquely as a finite join of pairwise incomparable, finitely join irreducible elements of L. By carefully generalizing our notions to partially ordered sets we get a similar theorem in this context.