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In algebra and logic, a modal algebra is a structure such that * is a Boolean algebra, * is a unary operation on A satisfying and for all x, y in A. Modal algebras provide models of propositional modal logics in the same way as Boolean algebras are models of classical logic. In particular, the variety of all modal algebras is the equivalent algebraic semantics of the modal logic K in the sense of abstract algebraic logic, and the lattice of its subvarieties is dually isomorphic to the lattice of normal modal logics.

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rdfs:label	Algèbre modale (fr) Modal algebra (en) 模態代數 (zh)
rdfs:comment	En algèbre et en logique, une algèbre modale est une structure tel que * est une algèbre de Boole (structure), * est un opérateur unaire sur A satisfaisant et pour tout x, y dans A. Une algèbre modale fourni des modèles de logiques modales propositionnel de la même manière que les algèbres booléennes sont des modèles de la logique classique. En particulier, la variété de toutes algèbres modales est la sémantique algébrique équivalentes de la logique modale K dans le sens de la , et le treillis de ses sous-variétés est duellement isomorphe au treillis de logiques modales normales. (fr) 在代數和邏輯中，模態代數是代數結構使得 * 是布爾代數， * 是在 A 上的一元運算，對於所有 A 中的 x, y 滿足和。模態代數提供了命題模態邏輯的模型，以和布爾代數是經典邏輯的模型相同的方式。特別是，所有模態代數的是在抽象代數邏輯意義下的模態邏輯 K 的等價代數語義，并且它的子簇們的格對偶同構於正規模態邏輯的格。 Stone布爾代數表示定理可以推廣為，它確保了每個模態代數可以為在模態內可容納的集合們的代數。 (zh) In algebra and logic, a modal algebra is a structure such that * is a Boolean algebra, * is a unary operation on A satisfying and for all x, y in A. Modal algebras provide models of propositional modal logics in the same way as Boolean algebras are models of classical logic. In particular, the variety of all modal algebras is the equivalent algebraic semantics of the modal logic K in the sense of abstract algebraic logic, and the lattice of its subvarieties is dually isomorphic to the lattice of normal modal logics. (en)
dcterms:subject	Boolean algebra Modal logic
Wikipage page ID	17257316 (xsd:integer)
Wikipage revision ID	978799819 (xsd:integer)
Link from a Wikipage to another Wikipage	Modal logic Interior algebra General frame Logic Propositional logic Lattice (order) Representation theorem Boolean algebra Heyting algebra Isomorphic Abstract algebra Abstract algebraic logic Modal logic Provability logic Boolean algebra (structure) Classical logic Variety (universal algebra) Normal modal logic Jónsson–Tarski duality Stone's representation theorem
sameAs	Modal algebra Modal algebra Modal algebra Modal algebra Modal algebra
dbp:wikiPageUsesTemplate	dbt:Algebra-stub dbt:ISBN
has abstract	In algebra and logic, a modal algebra is a structure such that * is a Boolean algebra, * is a unary operation on A satisfying and for all x, y in A. Modal algebras provide models of propositional modal logics in the same way as Boolean algebras are models of classical logic. In particular, the variety of all modal algebras is the equivalent algebraic semantics of the modal logic K in the sense of abstract algebraic logic, and the lattice of its subvarieties is dually isomorphic to the lattice of normal modal logics. Stone's representation theorem can be generalized to the Jónsson–Tarski duality, which ensures that each modal algebra can be represented as the algebra of admissible sets in a modal general frame. A Magari algebra (or diagonalizable algebra) is a modal algebra satisfying . Magari algebras correspond to provability logic. (en) En algèbre et en logique, une algèbre modale est une structure tel que * est une algèbre de Boole (structure), * est un opérateur unaire sur A satisfaisant et pour tout x, y dans A. Une algèbre modale fourni des modèles de logiques modales propositionnel de la même manière que les algèbres booléennes sont des modèles de la logique classique. En particulier, la variété de toutes algèbres modales est la sémantique algébrique équivalentes de la logique modale K dans le sens de la , et le treillis de ses sous-variétés est duellement isomorphe au treillis de logiques modales normales. (fr) 在代數和邏輯中，模態代數是代數結構使得 * 是布爾代數， * 是在 A 上的一元運算，對於所有 A 中的 x, y 滿足和。模態代數提供了命題模態邏輯的模型，以和布爾代數是經典邏輯的模型相同的方式。特別是，所有模態代數的是在抽象代數邏輯意義下的模態邏輯 K 的等價代數語義，并且它的子簇們的格對偶同構於正規模態邏輯的格。 Stone布爾代數表示定理可以推廣為，它確保了每個模態代數可以為在模態內可容納的集合們的代數。 (zh)
prov:wasDerivedFrom	wikipedia-en:Modal_algebra?oldid=978799819&ns=0
page length (characters) of wiki page	1584 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Modal_algebra
is Link from a Wikipage to another Wikipage of	Modal logic Unification (computer science) Interior algebra Wim Blok Alexandrov topology General frame Kripke semantics Admissible rule Algebraic logic Abstract algebraic logic Outline of logic
is foaf:primaryTopic of	wikipedia-en:Modal_algebra

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