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- Duality_theory_for_distributive_lattices abstract "In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This generalizes the well-known Stone duality between Stone spaces and Boolean algebras.Let L be a bounded distributive lattice, and let X denote the set of prime filters of L. For each a ∈ L, let φ+(a) = {x∈ X : a ∈ x} . Then (X,τ+) is a spectral space, where the topology τ+ on X is generated by {φ+(a) : a∈ L} . The spectral space (X,τ+) is called the prime spectrum of L.The map φ+ is a lattice isomorphism from L onto the lattice of all compact open subsets of (X,τ+). In fact, each spectral space is homeomorphic to the prime spectrum of some bounded distributive lattice.Similarly, if φ−(a) = {x∈ X : a ∉ x} and τ− denotes the topology generated by {φ−(a) : a∈ L}, then (X,τ−) is also a spectral space. Moreover, (X,τ+,τ−) is a pairwise Stone space. The pairwise Stone space (X,τ+,τ−) is called the bitopological dual of L. Each pairwise Stone space is bi-homeomorphic to the bitopological dual of some bounded distributive lattice.Finally, let ≤ be set-theoretic inclusion on the set of prime filters of L and let τ = τ+∨ τ−. Then (X,τ,≤) is a Priestley space. Moreover, φ+ is a lattice isomorphism from L onto the lattice of all clopen up-sets of (X,τ,≤). The Priestley space (X,τ,≤) is called the Priestley dual of L. Each Priestley space is isomorphic to the Priestley dual of some bounded distributive lattice.Let Dist denote the category of bounded distributive lattices and bounded lattice homomorphisms. Then the above three representations of bounded distributive lattices can be extended to dual equivalence between Dist and the categories Spec, PStone, and Pries of spectral spaces with spectral maps, of pairwise Stone spaces with bi-continuous maps, and of Priestley spaces with Priestley morphisms, respectively:Thus, there are three equivalent ways of representing bounded distributive lattices. Each one has its own motivation and advantages, but ultimately they all serve the same purpose of providing better understanding of bounded distributive lattices.".
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- Duality_theory_for_distributive_lattices wikiPageRevisionID "607146391".
- Duality_theory_for_distributive_lattices wikiPageWikiLink Bi-homeomorphic.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Birkhoffs_representation_theorem.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Boolean_algebra_(structure).
- Duality_theory_for_distributive_lattices wikiPageWikiLink Category:Category_theory.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Category:Lattice_theory.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Category:Topology.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Clopen_set.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Compact_set.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Compact_space.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Distributive_lattice.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Equivalence_of_categories.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Esakia_duality.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Homeomorphism.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Homomorphism.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Homomorphisms.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Ideal_(order_theory).
- Duality_theory_for_distributive_lattices wikiPageWikiLink Isomorphism.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Map_(mathematics).
- Duality_theory_for_distributive_lattices wikiPageWikiLink Mathematics.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Open_set.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Pairwise_Stone_space.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Priestley_space.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Set_(mathematics).
- Duality_theory_for_distributive_lattices wikiPageWikiLink Spectral_space.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Stone_duality.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Stone_space.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Stones_representation_theorem_for_Boolean_algebras.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Topological_space.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Up-set.
- Duality_theory_for_distributive_lattices wikiPageWikiLink Upper_set.
- Duality_theory_for_distributive_lattices wikiPageWikiLink File:DL_Duality.png.
- Duality_theory_for_distributive_lattices wikiPageWikiLinkText "Duality theory for distributive lattices".
- Duality_theory_for_distributive_lattices wikiPageWikiLinkText "duality theory for distributive lattices".
- Duality_theory_for_distributive_lattices wikiPageWikiLinkText "duality".
- Duality_theory_for_distributive_lattices wikiPageWikiLinkText "spectrum".
- Duality_theory_for_distributive_lattices hasPhotoCollection Duality_theory_for_distributive_lattices.
- Duality_theory_for_distributive_lattices wikiPageUsesTemplate Template:Math.
- Duality_theory_for_distributive_lattices wikiPageUsesTemplate Template:Reflist.
- Duality_theory_for_distributive_lattices subject Category:Category_theory.
- Duality_theory_for_distributive_lattices subject Category:Lattice_theory.
- Duality_theory_for_distributive_lattices subject Category:Topology.
- Duality_theory_for_distributive_lattices type Field.
- Duality_theory_for_distributive_lattices type Function.
- Duality_theory_for_distributive_lattices comment "In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This generalizes the well-known Stone duality between Stone spaces and Boolean algebras.Let L be a bounded distributive lattice, and let X denote the set of prime filters of L. For each a ∈ L, let φ+(a) = {x∈ X : a ∈ x} .".
- Duality_theory_for_distributive_lattices label "Duality theory for distributive lattices".
- Duality_theory_for_distributive_lattices sameAs ორადობის_თეორია_დისტრიბუციული_მესრებისათვის.
- Duality_theory_for_distributive_lattices sameAs m.0bh7z7r.
- Duality_theory_for_distributive_lattices sameAs Q5310267.
- Duality_theory_for_distributive_lattices sameAs Q5310267.
- Duality_theory_for_distributive_lattices wasDerivedFrom Duality_theory_for_distributive_lattices?oldid=607146391.
- Duality_theory_for_distributive_lattices depiction DL_Duality.png.
- Duality_theory_for_distributive_lattices isPrimaryTopicOf Duality_theory_for_distributive_lattices.