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A non-empty subset I of a partially ordered set (P,≤) is an ideal, if the following conditions hold:
While this is the most general way to define an ideal for arbitrary posets, it was originally defined for lattices only. In this case, the following equivalent definition can be given: A non-empty subset I of a lattice (P,≤) is an ideal, iff it is a lower set that is closed under finite joins (suprema), i.e., for all x, y in I, we find that x'y is also in I.
The dual notion of an ideal, i.e. the concept obtained by reversing all ≤ and exchanging with , is a filter. The terms order ideal and order filter are sometimes used for arbitrary lower or upper sets. Wikipedia uses only "ideal/filter (of order theory)" and "lower/upper set" in order to avoid confusion.
An ideal or filter is said to be proper if it is not equal to the whole set P.
The smallest ideal that contains a given element p is a principal ideal and p is said to be a principal element of the ideal in this situation. The principal ideal p for a principal p is thus given by p = {x in P | x ≤ 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 filter is necessarily proper. For lattices, prime ideals can be characterized as follows:
A subset I of a lattice (P,≤) is a prime ideal, iff
It is easily checked that this indeed is equivalent to stating that P\I is a filter (which is then also prime, in the dual sense).
For a complete lattice the notion of a completely prime ideal is known. It is defined to be a proper ideal I with the additional property that, whenever the meet ( infinum ) 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 Zermelo-Fraenkel set theory. 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 which is a strictly greater set than I. Likewise, a filter F is maximal if it is proper and there is no proper filter which is strictly greater.
When a poset is a distributive lattice, maximal ideals and filters are necessarily prime, while the converse of this statement is false in general.
Maximal filters are sometimes called ultrafilters, but this terminology if often reserved for Boolean algebras, where a maximal filter (ideal) is a filter (ideal) that contains exactly one of the elements {a, ¬a}, for each element a of the Boolean algebra. In Boolean algebras, the terms prime ideal and maximal ideal coincide, as do the terms prime filter and maximal filter.
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 which 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.
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 algebraIn mathematics and computer science, Boolean algebras or Boolean lattices are algebraic structures which "capture the essence" of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and compleme, this theorem is called the Boolean prime ideal theoremIn mathematics, a number of so called prime ideal theorems for guaranteeing the existence of certain subsets of an abstract algebra can be stated. Among the most popular statements of this form is the Boolean prime ideal theorem which states that ideals i. 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.
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