Prime Filter

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.

Basic definitions

A subset of a partially ordered set $(P, \leq)$ is an ideal, if the following conditions hold:

# is non-empty,
# for every ''x'' in and ''y'' in ''P'', implies that ''y'' is in  ( is a lower set),
# for every ''x'', ''y'' in , there is some element ''z'' in , such that and  ( is a directed set).

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 subset of a lattice $(P, \leq)$ 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 , the element $x \vee y$ of ''P'' is also in .

The dual notion of an ideal, i.e., the concept obtained by reversing all ≤ and exchanging $\vee$ with $\wedge,$ is a filter.

Some authors use the term ideal to mean a lower set, i.e., they include only condition 2 above,, p. 100/ref> 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.

Frink ideals, pseudoideals and Doyle pseudoideals are different generalizations of the notion of a lattice ideal.

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 and ''p'' is said to be a of the ideal in this situation. The principal ideal $\downarrow p$ for a principal ''p'' is thus given by .

Prime ideals

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 . 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:

A subset of a lattice $(P, \leq)$ is a prime ideal, if and only if

# is a proper ideal of ''P'', and
# for all elements ''x'' and ''y'' of ''P'', $x \wedge y$ in implies that or .

It is easily checked that this is indeed equivalent to stating that $P \setminus I$ is a filter (which is then also prime, in the dual sense).

For a complete lattice the further notion of a is meaningful.
It is defined to be a proper ideal with the additional property that, whenever the meet (infimum) of some arbitrary set is in , some element of ''A'' is also in .
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.

Maximal ideals

An ideal is a if it is proper and there is no ''proper'' ideal ''J'' that is a strict superset set of . 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.

Maximal filters are sometimes called ultrafilters, but this terminology is often reserved for Boolean algebras, where a maximal filter (ideal) is a filter (ideal) that contains exactly one of the elements , 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 and a filter ''F'' such that is disjoint