The Ideal (order theory) reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Ideal (order theory)

In the mathematical theory of partial orders, an ideal is a special subset of a partially ordered set. 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.

Table of contents

1 Basic definitions 2 Prime ideals 3 Maximal ideals 4 Applications 5 History

Basic definitions

A non-empty subset I of a partially ordered set (P,≤) is an ideal, if the following conditions hold:

1. For every x in I, y ≤ x implies that y is in I. (I is a lower set)
2. For every x, y in I, there is some element z in I, such that x ≤ z and y ≤ z. (I is a directed set)

While this is the most general way to define an ideal for arbitrary posets, it was originally defined for latticess 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 v y is also in I.

The dual notion of an ideal, i.e. the concept obtained by reversing all ≤ and exchanging ^ and v, 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 for p is just given by the set {x in P | x ≤ p} and is denoted by prefixing p with a downward arrow.

Prime ideals

An important special case of an ideal is given 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

1. I is an ideal of P, and
2. for every elements x and y of P, x ^ y in I implies that x is in P or y is in P.

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 know. It is defined to be a proper ideal I with the additional property that, whenever the join (supremum) of some arbitrary set A is in I, some element of A is also in I. So this is just a specific prime filter that extends the above conditions for infinite joins.

Maximal 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.

It can be shown that all maximal ideals are prime ideals, and that all maximal filters are prime filters.

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 this case, it can also be shown that M is a prime ideal.

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.

Applications

The construction of ideals and filters is an important tool in many applications of order theory.

• In Stone's representation theorem for Boolean algebras, the maximal ideals (or, equivalently via the negation map, ultrafilters) are used to obtain the set of points of a topological space, whose clopen sets are isomorphic to the original Boolean algebra.

• Order theory knows many completion procedures, to turn posets into posets with additional completeness properties. For example, the ideal completion of a given partial order P is the set of all ideals of P ordered by subset inclusion. This construction yields the free dcpo generated by P.

History

Ideals were introduced first by Marshall H. Stone, who derived their name from the ring ideals of abstract algebra. This teminology is due to the fact that, using the isomorphism of the categories of Boolean algebras and of Boolean rings, both notions do indeed coincide.

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