In the mathematical field of
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, an ideal is a
partially ordered
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. A poset consists of a set together with a binary r ...
collection of
sets that are considered to be "small" or "negligible". Every
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and the
union of any two elements of the ideal must also be in the ideal.
More formally, given a set
an ideal
on
is a
nonempty subset of the
powerset
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of
such that:
#
# if
and
then
and
# if
then
Some authors add a fourth condition that
itself is not in
; ideals with this extra property are called .
Ideals in the set-theoretic sense are exactly
ideals in the order-theoretic sense, where the relevant order is set inclusion. Also, they are exactly
ideals in the ring-theoretic sense on the
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 ...
formed by the powerset of the underlying set. The dual notion of an ideal 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 ...
.
Terminology
An element of an ideal
is said to be or , or simply or if the ideal
is understood from context. If
is an ideal on
then a subset of
is said to be (or just ) if it is an element of
The collection of all
-positive subsets of
is denoted
If
is a proper ideal on
and for every
either
or
then
is a .
Examples of ideals
General examples
* For any set
and any arbitrarily chosen subset
the subsets of
form an ideal on
For finite
all ideals are of this form.
* The
finite subsets of any set
form an ideal on
* For any
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
, subsets of sets of measure zero.
* For any
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
, sets of finite measure. This encompasses finite subsets (using
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ...
) and small sets below.
* A
bornology
In mathematics, especially functional analysis, a bornology on a set ''X'' is a collection of subsets of ''X'' satisfying axioms that generalize the notion of boundedness. One of the key motivations behind bornologies and bornological analysis is ...
on a set
is an ideal that
covers
* A non-empty family
of subsets of
is a proper ideal on
if and only if its in
which is denoted and defined by
is a proper
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 ...
on
(a filter is if it is not equal to
). The dual of the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
is itself; that is,
Thus a non-empty family
is an ideal on
if and only if its dual
is a
dual ideal on
(which by definition is either the power set
or else a proper filter on
).
Ideals on the natural numbers
* The ideal of all finite sets of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s is denoted Fin.
* The on the natural numbers, denoted
is the collection of all sets
of natural numbers such that the sum
is finite. See
small set.
* The on the natural numbers, denoted
is the collection of all sets
of natural numbers such that the fraction of natural numbers less than
that belong to
tends to zero as
tends to infinity. (That is, the
asymptotic density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desi ...
of
is zero.)
Ideals on the real numbers
* The is the collection of all sets
of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s such that the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
of
is zero.
* The is the collection of all
meager set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
s of real numbers.
Ideals on other sets
* If
is an
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the leas ...
of uncountable
cofinality
In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''.
This definition of cofinality relies on the axiom of choice, as it uses t ...
, the on
is the collection of all subsets of
that are not
stationary set In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in measure theory. There are at least three cl ...
s. This ideal has been studied extensively by
W. Hugh Woodin
William Hugh Woodin (born April 23, 1955) is an American mathematician and set theorist at Harvard University. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinals, the Woodin cardinals, ...
.
Operations on ideals
Given ideals
and
on underlying sets
and
respectively, one forms the product
on the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
as follows: For any subset
That is, a set is negligible in the product ideal if only a negligible collection of
-coordinates correspond to a non-negligible slice of
in the
-direction. (Perhaps clearer: A set is in the product ideal if positively many
-coordinates correspond to positive slices.)
An ideal
on a set
induces an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
on
the powerset of
considering
and
to be equivalent (for
subsets of
) if and only if the
symmetric difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \.
Th ...
of
and
is an element of
The
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of
by this equivalence relation 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 i ...
, denoted
(read "P of
mod
").
To every ideal there is a corresponding
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 ...
, called its . If
is an ideal on
then the dual filter of
is the collection of all sets
where
is an element of
(Here
denotes the
relative complement
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in .
When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is ...
of
in
; that is, the collection of all elements of
that are in
).
Relationships among ideals
If
and
are ideals on
and
respectively,
and
are if they are the same ideal except for renaming of the elements of their underlying sets (ignoring negligible sets). More formally, the requirement is that there be sets
and
elements of
and
respectively, and a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
such that for any subset
if and only if the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of
under
If
and
are Rudin–Keisler isomorphic, then
and
are isomorphic as Boolean algebras. Isomorphisms of quotient Boolean algebras induced by Rudin–Keisler isomorphisms of ideals are called .
See also
*
*
*
*
*
*
*
References
* {{cite book, last=Farah, first=Ilijas, series=Memoirs of the AMS, publisher=American Mathematical Society, date=November 2000, title=Analytic quotients: Theory of liftings for quotients over analytic ideals on the integers, isbn=9780821821176, url=https://books.google.com/books?id=IP7TCQAAQBAJ&q=ideal+OR+ideals
Set theory