Ideal (order theory)
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.
Some authors use the term ideal to mean a lower set, i.e., they include only condition 2 above, while others use the term order ideal for this weaker notion. With the weaker definition, an ideal of a lattice seen as a poset is not closed under joins, so it is not necessarily an ideal of the lattice. Wikipedia uses only "ideal/filter (of order theory)" and "lower/upper set" to avoid confusion.
An ideal or filter is said to be proper if it is not equal to the whole set P.
An important special case of an ideal is constituted by those ideals whose set-theoretic complements are filters, i.e. ideals in the inverse order. Such ideals are called prime ideals. Also note that, since we require ideals and filters to be non-empty, every prime ideal is necessarily proper. For lattices, prime ideals can be characterized as follows:
It is easily checked that this is indeed equivalent to stating that P \ I is a filter (which is then also prime, in the dual sense).
For a complete lattice the further notion of a completely prime ideal is meaningful. 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.
The existence of prime ideals is in general not obvious, and often a satisfactory amount of prime ideals cannot be derived within ZF (Zermelo–Fraenkel set theory without the axiom of choice). This issue is discussed in various prime ideal theorems, which are necessary for many applications that require prime ideals.
An ideal I is maximal if it is proper and there is no proper ideal J that is a strict superset set of I. Likewise, a filter F is maximal if it is proper and there is no proper filter that is a strict superset.
When a poset is a distributive lattice, maximal ideals and filters are necessarily prime, while the converse of this statement is false in general.
There is another interesting notion of maximality of ideals: Consider an ideal I and a filter F such that I is disjoint from F. We are interested in an ideal M that is maximal among all ideals that contain I and are disjoint from F. In the case of distributive lattices such an M is always a prime ideal. A proof of this statement follows.
For the other case, assume that there is some m in M with m ∨ a in F. Now if any element n in M is such that n ∨ b is in F, one finds that (m ∨ n) ∨ b and (m ∨ n) ∨ a are both in F. But then their meet is in F and, by distributivity, (m ∨ n) ∨ (a ∧ b) is in F too. 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.
However, in general it is not clear whether there exists any ideal M that is maximal in this sense. 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. In the special case that the considered order is a Boolean algebra, this theorem is called the Boolean prime ideal theorem. 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.
Ideals were introduced first by Marshall H. Stone, who derived their name from the ring ideals of abstract algebra. He adopted this terminology because, using the isomorphism of the categories of Boolean algebras and of Boolean rings, the two notions do indeed coincide.