NationMaster - Encyclopedia: 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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ...

1 Basic definitions
2 Prime ideals
3 Maximal ideals
4 Applications
5 History
6 Literature
7 See also

Basic definitions

A non-empty subset I of a partially ordered set (P,≤) is an ideal, if the following conditions hold: In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ...

For every x in I, y ≤ x implies that y is in I. (I is a lower set)
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 lattices only. In this case, the following equivalent definition can be given: A non-empty subset I of a lattice (P,≤) is an ideal, if and only if it is a lower set that is closed under finite joins (suprema), i.e., for all x, y in I, we find that x $vee$ y is also in I. In mathematics, an upper set, or upward closed set, is a subset Y of a given partially ordered set (X,â‰¤) such that, for all elements x and y, if x is less than or equal to y and x is an element of Y, then y is also in Y... In mathematics, a directed set is a set A together with a binary relation ≤ having the following properties: a ≤ a for all a in A (reflexivity) if a ≤ b and b ≤ c, then a ≤ c (transitivity) for any two a and b in A, there... The name lattice is suggested by the form of the Hasse diagram depicting it. ... â†” â‡” â‰¡ logical symbols representing iff. ... In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ...

The dual notion of an ideal, i.e. the concept obtained by reversing all ≤ and exchanging $vee$ with $wedge$ , 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. In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ... In mathematics, a filter is a special subset of a partially ordered set. ...

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 $downarrow$ p for a principal p is thus given by $downarrow$ p = {x in P | x ≤ p}.

Prime ideals

Main article: Prime ideal

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: In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...

A subset I of a lattice (P,≤) is a prime ideal, if and only if

I is an ideal of P, and
for every elements x and y of P, x $wedge$ y in I implies that x is in I or y is in I.

It is easily checked that this indeed is equivalent to stating that PI 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 (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. In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. ...

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. Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ... In mathematics, a number of prime ideal theorems for guaranteeing the existence of certain subsets of an abstract algebra can be stated. ...

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.

When a poset is a distributive lattice, maximal ideals and filters are necessarily prime, while the converse of this statement is false in general. In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. ...

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 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. In mathematics, two sets are said to be disjoint if they have no element in common. ...

Proof. Assume the ideal M is maximal with respect to disjointness from the filter F. Suppose for a contradiction that M is not prime, i.e. there exists a pair of elements a and b such that a $wedge$ b in M but neither a nor b are in M. Consider the case that for all m in M, m $vee$ a is not in F. One can construct an ideal N by taking the downward closure of the set of all binary joins of this form, i.e. N = { x | x≤ m $vee$ a for some m in M}. It is readily checked that N is indeed an ideal disjoint from F which is strictly greater than M. But this contradicts the maximality of M and thus the assumption that M is not prime.

For the other case, assume that there is some m in M with m $vee$ a in F. Now if any element n in M is such that n $vee$ b is in F, one finds that (m $vee$ n) $vee$ b and (m $vee$ n) $vee$ a are both in F. But then their meet is in F and, by distributivity, (m $vee$ n) $vee$ (a $wedge$ 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. In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... In mathematics, a number of prime ideal theorems for guaranteeing the existence of certain subsets of an abstract algebra can be stated. ...

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. Furthermore the ideal completion serves to reconstruct any algebraic dcpo from its set of compact elements.

In mathematics, Stones representation theorem for Boolean algebras, named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Stone spaces, i. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, a clopen set (or closed-open set, a portmanteau word) in a topological space is a set which is both open and closed. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x â‰¤ a implies that x is in A as well) and B is closed upwards... In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set. ... In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. ... In mathematics, directed complete partial orders and complete partial orders are special classes of partially ordered sets. ... In the mathematical area of order theory, the compact or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element. ... In the mathematical area of order theory, the compact or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any directed set that does not already contain members above the compact element. ...

History

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, both notions do indeed coincide. Marshall Harvey Stone (April 8, 1903–January 9, 1989) was a American mathematician who made several important contributions in various areas of mathematical analysis, including in particular functional analysis. ... In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i. ... In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists of idempotent elements. ...

Literature

Ideals and filters are among the most basic concepts of order theory. See the introductory books given for order theory and lattice theory, and the literature on the Boolean prime ideal theorem. Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... In mathematics, a number of prime ideal theorems for guaranteeing the existence of certain subsets of an abstract algebra can be stated. ...

A monograph available free online:

Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.

Contents

Prime ideals

Maximal ideals

Applications

History

Literature

See also