Birkhoff’s representation theorem for finite distributive lattices
E637942
Birkhoff’s representation theorem for finite distributive lattices is a fundamental result in lattice theory that characterizes every finite distributive lattice as isomorphic to the lattice of lower (order) ideals of a finite poset.
Statements (31)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
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result in lattice theory ⓘ |
| appliesTo | finite distributive lattices ⓘ |
| characterizes | finite distributive lattices ⓘ |
| coreIdea | algebraic structures can be represented as lattices of sets with inclusion order ⓘ |
| describesAsIsomorphic |
finite distributive lattice
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lattice of lower ideals of a finite poset ⓘ |
| field |
lattice theory
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order theory ⓘ |
| generalizationOf | representation of Boolean algebras as fields of sets ⓘ |
| hasConsequence | finite distributive lattices are completely determined by their posets of join-irreducible elements ⓘ |
| hasVariant | Birkhoff’s representation theorem for arbitrary distributive lattices using spectral spaces NERFINISHED ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| implies | classification of finite distributive lattices up to isomorphism by finite posets up to isomorphism ⓘ |
| involvesConcept |
distributive lattice
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finite lattice ⓘ lattice isomorphism ⓘ lower set ⓘ order ideal ⓘ partially ordered set ⓘ |
| namedAfter | Garrett Birkhoff NERFINISHED ⓘ |
| providesRepresentationOf | finite distributive lattices by posets ⓘ |
| relates |
finite distributive lattices
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finite posets ⓘ |
| statesThat |
every finite distributive lattice is isomorphic to the lattice of lower ideals of a finite poset
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for every finite distributive lattice there exists a finite poset whose lattice of order ideals is isomorphic to it ⓘ |
| usedIn |
combinatorics
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theory of posets ⓘ universal algebra NERFINISHED ⓘ |
| usesConstruction |
lattice of order ideals of a poset
ⓘ
set of all lower sets of a finite poset ordered by inclusion ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
Garrett Birkhoff
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Birkhoff’s representation theorem for finite distributive lattices
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