Decompositions In Complete Supper Principal Divisor Lattices

This paper deals with the decompositions of elements in complete supper principal divisor lattices. Firstly, a concept of supper principal divisor lattice is introduced and the conclusion that every complete, supper principal divisor, distributive lattice is a modular lattice is proved. Secondly, by investigating the structure of weakly atomic, supper principal divisor lattice, we can easily come to the conclusion that every complete, supper principal divisor, weakly atomic lattice is a lower semimodular lattice. Under this condition, one necessary and sufficient condition about elements decomposition is given and some proposition about decompositions of elements is shown as well. Furthermore, by introducing the concept of locally, lower-semimodular lattice, we prove that in every complete, supper principal divisor, weakly atomic lattice L, every supper principal divisor element has a replaceable, complete join-irredundant decomposition if and only if L is a locally, lower-semimodular lattice. What:more, the conclusion that every compactly generated, locally modular lattice is a lower-semimodular lattice is obtained. These results partly solve the problem raised by Dilworth.