How can i define finite direct decomposition of elements and indecomposable elements at arbitrary lattice .
I think i can say an element of lattice is finite direct decomposition of elements if it be written as a maximal of some finite elements .For example if we look at lattice of sets i can say a set is finite direct composition of elements if it can be written as a union of finite set . Am i right ? i search on google but i can find nothing any comment or link will be helpful Thanks.
I'm not shure, if you are right. But if I'm right, you are looking for reducible and irreducible elements both exist in weaker versions: supremum irreducible elements and infimum irreducible elements. You will probably not find too much about the finitely reducible case: It is trivial for a (semi-)lattice. If you want to understand this fact, you could prove as an exercise that every finite lattice is a complete lattice.