In
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, an ideal is a special subset of a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
(poset). Although this term historically was derived from the notion of a
ring ideal
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers pre ...
of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and
lattice theory
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper boun ...
.
Basic definitions
A subset of a partially ordered set
is an ideal, if the following conditions hold:
# is
non-empty
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...
,
# for every ''x'' in and ''y'' in ''P'', implies that ''y'' is in ( is a
lower set
In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger ...
),
# for every ''x'', ''y'' in , there is some element ''z'' in , such that and ( is a
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
).
While this is the most general way to define an ideal for arbitrary posets, it was originally defined for
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
s only. In this case, the following equivalent definition can be given:
a subset of a lattice
is an ideal
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
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
of ''P'' is also in .
The
dual notion of an ideal, i.e., the concept obtained by reversing all ≤ and exchanging
with
is a
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
.
Some authors use the term ideal to mean a lower set, i.e., they include only condition 2 above,
[, ]p. 100 P. is an abbreviation or acronym that may refer to:
* Page (paper), where the abbreviation comes from Latin ''pagina''
* Paris Herbarium, at the ''Muséum national d'histoire naturelle''
* ''Pani'' (Polish), translating as Mrs.
* The ''Pacific Repo ...
/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 ideal In mathematics, a Frink ideal, introduced by Orrin Frink, is a certain kind of subset of a partially ordered set.
Basic definitions
LU(''A'') is the set of all common lower bounds of the set of all common upper bounds of the subset ''A'' of a par ...
s, pseudoideal In the theory of partially ordered sets, a pseudoideal is a subset characterized by a bounding operator LU. Basic definitions
LU(''A'') is the set of all lower bounds of the set of all upper bounds of the subset ''A'' of a partially ordered set.
...
s 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 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 is a prime ideal, if and only if
# is a proper ideal of ''P'', and
# for all elements ''x'' and ''y'' of ''P'', in implies that or .
It is easily checked that this is indeed equivalent to stating that is a filter (which is then also prime, in the dual sense).
For a complete lattice
In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally 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
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
) 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
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
without the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
).
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
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set uni ...
, maximal ideals and filters are necessarily prime, while the converse of this statement is false in general.
Maximal filters are sometimes called ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on ...
s, 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 from ''F''. We are interested in an ideal ''M'' that is maximal among all ideals that contain 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 mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
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
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
, this theorem is called the Boolean prime ideal theorem
In mathematics, the Boolean prime ideal theorem states that Ideal (order theory), ideals in a Boolean algebra (structure), Boolean algebra can be extended to Ideal (order theory)#Prime ideals , prime ideals. A variation of this statement for Filt ...
. 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.
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
In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first hal ...
, the maximal ideals (or, equivalently via the negation map, ultrafilters) are used to obtain the set of points of a topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
, whose clopen set
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical d ...
s are isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
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
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
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.
History
Ideals were introduced first by Marshall H. Stone
Marshall Harvey Stone (April 8, 1903 – January 9, 1989) was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras.
Biography
Stone was the son of Harlan Fiske Stone, who wa ...
, 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 ring In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2.
Every Boolean ring gives rise to a Boolean al ...
s, the two notions do indeed coincide.
Literature
Ideals and filters are among the most basic concepts of order theory. See the introductory books given for order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
and lattice theory
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper boun ...
, and the literature on the Boolean prime ideal theorem
In mathematics, the Boolean prime ideal theorem states that Ideal (order theory), ideals in a Boolean algebra (structure), Boolean algebra can be extended to Ideal (order theory)#Prime ideals , prime ideals. A variation of this statement for Filt ...
.
See also
*
*
*
Notes
References
*
*
*
*
*
{{Order theory
Articles containing proofs
Ideals (ring theory)
Order theory