Lattice (order)
Important lattice-theoretic notions

We now define some order-theoretic notions of importance to lattice theory. In the following, let x {\displaystyle x} be an element of some lattice L. {\displaystyle L.} x {\displaystyle x} is called:

Let L {\displaystyle L} have a bottom element 0. An element x {\displaystyle x} of L {\displaystyle L} is an atom if 0 < x {\displaystyle 0<x} and there exists no element y ∈ L {\displaystyle y\in L} such that 0 < y < x. {\displaystyle 0<y<x.} Then L {\displaystyle L} is called:

However, many sources and mathematical communities use the term "atomic" to mean "atomistic" as defined above.

The notions of ideals and the dual notion of filters refer to particular kinds of subsets of a partially ordered set, and are therefore important for lattice theory. Details can be found in the respective entries.
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