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In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y which admits operations x\z and z/y, loosely analogous to division or implication, when x•y is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations coincide when the monoid is commutative. The general concept was introduced by Morgan Ward and Robert P. Dilworth in 1939. Examples, some of which existed prior to the general concept, include Boolean algebras, Heyting algebras, residuated Boolean algebras, relation algebras, and MV-algebras. Residuated semilattices omit the meet operation ∧, for example Kleene algebras and action algebras.

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rdfs:label	Residuated lattice (en) Ґратка з діленням (uk) 剩余格 (zh)
rdfs:comment	In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y which admits operations x\z and z/y, loosely analogous to division or implication, when x•y is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations coincide when the monoid is commutative. The general concept was introduced by Morgan Ward and Robert P. Dilworth in 1939. Examples, some of which existed prior to the general concept, include Boolean algebras, Heyting algebras, residuated Boolean algebras, relation algebras, and MV-algebras. Residuated semilattices omit the meet operation ∧, for example Kleene algebras and action algebras. (en) 在抽象代数中，剩余格是既为格又为幺半群的代数结构，使得幺半群乘法的每个自变量都是关于这个格次序的伽罗瓦连接的一极。它的一般概念是Ward和Dilworth在1939年介入的。某些例子先于一般概念而存在，包括布尔代数、Heyting代数、剩余布尔代数、关系代数和MV-代数。剩余半格省略了交运算∧，比如克莱尼代数和作用代数。 (zh) Ґратка з діленням — алгебраїчна структура в теорії ґраток, що одночасно є ґраткою x ≤ y та моноїдом x•y, яка дозволяє операції x\z та z/y, що є аналогами ділення чи імплікації, якщо розглядати x•y як множення чи кон'юнкцію, відповідно. Прикладами ґраток з діленням є булеві алгебри, , алгебри Гейтінга, . (uk)
dcterms:subject	Lattice theory Fuzzy logic Mathematical logic Ordered algebraic structures
Wikipage page ID	2483542 (xsd:integer)
Wikipage revision ID	1104495893 (xsd:integer)
Link from a Wikipage to another Wikipage	MV-algebra Monus Morgan Ward Binary relations Algebraic structure Peirce's law Lattice theory Mathematics Residuated mapping Quantale Galois connection Monoid Conductor (ring theory) Trans. Amer. Math. Soc. Annihilator (ring theory) Linear logic Commutative algebra Ideal (ring theory) Action algebra Fuzzy logic Total order Distributive lattice Lattice (order) Formal language Robert P. Dilworth Relation algebra Relevance logic Ring (mathematics) Heyting algebra Heyting algebras Tautology (logic) Abstract algebra Mathematical logic Ordered algebraic structures Boolean algebra (structure) Kleene algebra Variety (universal algebra) Natural language Residuated Boolean algebra Residuated lattice Substructural logic
sameAs	Residuated lattice Residuated lattice Residuated lattice Residuated lattice Residuated lattice Residuated lattice
dbp:wikiPageUsesTemplate	dbt:Cn dbt:Isbn dbt:Relevance-inline
has abstract	In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y which admits operations x\z and z/y, loosely analogous to division or implication, when x•y is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations coincide when the monoid is commutative. The general concept was introduced by Morgan Ward and Robert P. Dilworth in 1939. Examples, some of which existed prior to the general concept, include Boolean algebras, Heyting algebras, residuated Boolean algebras, relation algebras, and MV-algebras. Residuated semilattices omit the meet operation ∧, for example Kleene algebras and action algebras. (en) 在抽象代数中，剩余格是既为格又为幺半群的代数结构，使得幺半群乘法的每个自变量都是关于这个格次序的伽罗瓦连接的一极。它的一般概念是Ward和Dilworth在1939年介入的。某些例子先于一般概念而存在，包括布尔代数、Heyting代数、剩余布尔代数、关系代数和MV-代数。剩余半格省略了交运算∧，比如克莱尼代数和作用代数。 (zh) Ґратка з діленням — алгебраїчна структура в теорії ґраток, що одночасно є ґраткою x ≤ y та моноїдом x•y, яка дозволяє операції x\z та z/y, що є аналогами ділення чи імплікації, якщо розглядати x•y як множення чи кон'юнкцію, відповідно. Прикладами ґраток з діленням є булеві алгебри, , алгебри Гейтінга, . (uk)
gold:hypernym	Structure
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foaf:isPrimaryTopicOf	wikipedia-en:Residuated_lattice
is Link from a Wikipage to another Wikipage of	MV-algebra Morgan Ward Dedekind–MacNeille completion Residuated mapping Quantale Idealizer Map of lattices Fuzzy logic T-norm Relevance logic Residual Heyting algebra BL (logic) Bunched logic Monoidal t-norm logic Residuated Boolean algebra Residuated lattice Multi-adjoint logic programming T-norm fuzzy logics Substructural logic Residuated semilattice
is Wikipage redirect of	Residuated semilattice

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