# Definition of finite direct decomposition of elements and indecomposable elements at arbitrary lattice

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.

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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.

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i edit question it is about decomposition and indecomposable elements at lattice as modules –  rmznyzgyr Mar 23 '14 at 21:43
The answer is still the same: The keyword you must use for search engines is called “irreducible” in lattice theory (lattice as in lattice order), at least if you didn't mix up different lattice notions in your set example. If this doen't help, you should provide the definitions of the notions you use. What is a lattice? What are the lattice operations? Which order are you using? If I'm mistaken, you should also retag your question, remove order-theory and lattice-orders and add group-theory or module-theory. BTW every set is the union of finite sets (the singletons of its elements). –  Keinstein Mar 23 '14 at 23:26