Ideals are of great importance for many constructions in order and lattice theory.
is an ideal if and only if it is a lower set that is closed under finite joins (suprema); that is, it is nonempty and for all x, y in I, the element
[3] A weaker notion of order ideal is defined to be a subset of a poset P that satisfies the above conditions 1 and 2.
The dual notion of an ideal, i.e., the concept obtained by reversing all ≤ and exchanging
It is defined to be a proper ideal I with the additional property that, whenever the meet (infimum) of some arbitrary set A is in I, some element of A is also in I.
So this is just a specific prime ideal that extends the above conditions to infinite meets.
When a poset is a distributive lattice, maximal ideals and filters are necessarily prime, while the converse of this statement is false in general.
Assume the ideal M is maximal with respect to disjointness from the filter F. Suppose for a contradiction that M is not prime, i.e. there exists a pair of elements a and b such that a ∧ b in M but neither a nor b are in M. Consider the case that for all m in M, m ∨ a is not in F. One can construct an ideal N by taking the downward closure of the set of all binary joins of this form, i.e. N = { x | x ≤ m ∨ a for some m ∈ M}.
It is readily checked that N is indeed an ideal disjoint from F which is strictly greater than M. But this contradicts the maximality of M and thus the assumption that M is not prime.
On the other hand, this finite join of elements of M is clearly in M, such that the assumed existence of n contradicts the disjointness of the two sets.
Hence all elements n of M have a join with b that is not in F. Consequently one can apply the above construction with b in place of a to obtain an ideal that is strictly greater than M while being disjoint from F. This finishes the proof.
Yet, if we assume the axiom of choice in our set theory, then the existence of M for every disjoint filter–ideal-pair can be shown.
It is strictly weaker than the axiom of choice and it turns out that nothing more is needed for many order-theoretic applications of ideals.
The construction of ideals and filters is an important tool in many applications of order theory.