About: Ideal (set theory)

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In the mathematical field of set theory, an ideal is a partially ordered collection of sets that are considered to be "small" or "negligible". Every subset 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 of such that: 1. * 2. * if and then and 3. * if then

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rdf:type	book
rdfs:label	Ideal (set theory) (en)
rdfs:comment	In the mathematical field of set theory, an ideal is a partially ordered collection of sets that are considered to be "small" or "negligible". Every subset 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 of such that: 1. * 2. * if and then and 3. * if then (en)
dct:subject	Set theory
Wikipage page ID	12633368 (xsd:integer)
Wikipage revision ID	1112188284 (xsd:integer)
Link from a Wikipage to another Wikipage	Cartesian product Relative complement Meager set Quotient set Empty set Ideal (ring theory) Measure space Set theory W. Hugh Woodin Lebesgue measure Equivalence relation Filter (set theory) Finite set Counting measure Bijection Cofinality Small set (combinatorics) Boolean algebra (structure) Boolean ring Ideal (order theory) Natural number Asymptotic density Ordinal number Real number Set (mathematics) Set theory Union (set theory) Symmetric difference Image (mathematics) Partial order Subset Stationary set Powerset
Link from a Wikipage to an external page	https://books.google.com/books%3Fid=IP7TCQAAQBAJ&q=ideal+OR+ideals
sameAs	Ideal (set theory) Ideal (set theory) Ideal (set theory) Ideal (set theory)
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has abstract	In the mathematical field of set theory, an ideal is a partially ordered collection of sets that are considered to be "small" or "negligible". Every subset 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 of such that: 1. * 2. * if and then and 3. * if then Some authors add a fourth condition that itself is not in ; ideals with this extra property are called proper ideals. 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 formed by the powerset of the underlying set. The dual notion of an ideal is a filter. (en)
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foaf:isPrimaryTopicOf	wikipedia-en:Ideal_(set_theory)
is Link from a Wikipage to another Wikipage of	Random algebra Measure (mathematics) Negligible set Sigma-ideal Baby Modula-3 Filter (set theory) Filters in topology Cardinal characteristic of the continuum Cardinal function Cichoń's diagram BRST quantization Set theory of the real line Large set (combinatorics) Blackboard bold Small set Martín Abadi Ideal (order theory) Ideal Ramsey cardinal Reflecting cardinal Riemann series theorem Dual filter Ideal of sets Positive set
is Wikipage redirect of	Dual filter Ideal of sets Positive set
is Wikipage disambiguates of	Ideal
is foaf:primaryTopic of	wikipedia-en:Ideal_(set_theory)

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