Lattice (order) - Important Lattice-theoretic Notions

Important Lattice-theoretic Notions

We now define some order-theoretic notions of importance to lattice theory. In the following, let x be an element of some lattice L. If L has a 0, x≠0 is sometimes required. x is:

• Join irreducible iff x = a∨b implies x = a or x = b for any a,b in L. When the first condition is generalized to arbitrary joins, x is called completely join irreducible (or ∨-irreducible). The dual notion is meet irreducibility (∧-irreducible);
• Join prime iff x ≤ a ∨ b implies x ≤ a or x ≤ b. This too can be generalized to obtain the notion completely join prime. The dual notion is meet prime. Any join-prime element is also join irreducible, and any meet-prime element is also meet irreducible. The converse holds if L is distributive.

Let L have a 0. An element x of L is an atom if 0 < x and there exists no element y of L such that 0 < y < x. We then say that L is:

• Atomic if for every nonzero element x of L, there exists an atom a of L such that a ≤ x ;
• Atomistic if every element of L is a supremum of atoms. That is, for all a, b in L such that there exists an atom x of L such that and

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