Template:Short description Template:More footnotes 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.
DefinitionsEdit
A subset Template:Mvar of a partially ordered set <math>(P, \leq)</math> is an ideal, if the following conditions hold:<ref>Template:Harvtxt, [[[:Template:Google books]] p. 141]: "A directed lower subset of a poset X is called an ideal"</ref><ref>Template:Cite book</ref>
- Template:Mvar is non-empty,
- for every x in Template:Mvar and y in P, Template:Math implies that y is in Template:Mvar (Template:Mvar is a lower set),
- for every x, y in Template:Mvar, there is some element z in Template:Mvar, such that Template:Math and Template:Math (Template:Mvar 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 Template:Mvar of a lattice <math>(P, \leq)</math> 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 Template:Mvar, the element <math>x \vee y</math> of P is also in Template:Mvar.Template:Sfn
Template:AnchorA weaker notion of order ideal is defined to be a subset of a poset Template:Mvar that satisfies the above conditions 1 and 2. In other words, an order ideal is simply a lower set. Similarly, an ideal can also be defined as a "directed lower set".
The dual notion of an ideal, i.e., the concept obtained by reversing all ≤ and exchanging <math>\vee</math> with <math>\wedge,</math> is a filter.
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.Template:Sfn
The smallest ideal that contains a given element p is a Template:Em and p is said to be a Template:Em of the ideal in this situation. The principal ideal <math>\downarrow p</math> for a principal p is thus given by Template:Math.
Terminology confusionEdit
The above definitions of "ideal" and "order ideal" are the standard ones, Template:SfnTemplate:SfnTemplate:Sfn but there is some confusion in terminology. Sometimes the words and definitions such as "ideal", "order ideal", "Frink ideal", or "partial order ideal" mean one another.<ref>Template:Citation</ref><ref>Template:Citation</ref>
Prime idealsEdit
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 Template:Visible anchors. 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 Template:Mvar of a lattice <math>(P, \leq)</math> is a prime ideal, if and only if
- Template:Mvar is a proper ideal of P, and
- for all elements x and y of P, <math>x \wedge y</math> in Template:Mvar implies that Template:Math or Template:Math.
It is easily checked that this is indeed equivalent to stating that <math>P \setminus I</math> is a filter (which is then also prime, in the dual sense).
For a complete lattice the further notion of a Template:Visible anchor is meaningful. It is defined to be a proper ideal Template:Mvar with the additional property that, whenever the meet (infimum) of some arbitrary set Template:Math is in Template:Math, some element of A is also in Template:Mvar. 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 idealsEdit
An ideal Template:Mvar is a Template:Em if it is proper and there is no proper ideal J that is a strict superset of Template:Mvar. 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 {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 Template:Mvar and a filter F such that Template:Mvar is disjoint from F. We are interested in an ideal M that is maximal among all ideals that contain Template:Mvar 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 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.
ApplicationsEdit
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. An ideal is principal if and only if it is compact in the ideal completion, so the original poset can be recovered as the sub-poset consisting of compact elements. Furthermore, every algebraic dcpo can be reconstructed as the ideal completion of its set of compact elements.
HistoryEdit
Ideals were introduced by Marshall H. Stone first for Boolean algebras,<ref>Template:Harvtxt and Template:Harvtxt</ref> where the name was derived 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.
Generalization to any posets was done by Frink.<ref>Template:Harvtxt</ref>
See alsoEdit
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- Semigroup ideal
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