Residuated Boolean algebra (original) (raw)
In mathematics, a residuated Boolean algebra is a residuated lattice whose lattice structure is that of a Boolean algebra. Examples include Boolean algebras with the monoid taken to be conjunction, the set of all formal languages over a given alphabet Σ under concatenation, the set of all binary relations on a given set X under relational composition, and more generally the power set of any equivalence relation, again under relational composition. The original application was to relation algebras as a finitely axiomatized generalization of the binary relation example, but there exist interesting examples of residuated Boolean algebras that are not relation algebras, such as the language example.
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| dbo:abstract | In mathematics, a residuated Boolean algebra is a residuated lattice whose lattice structure is that of a Boolean algebra. Examples include Boolean algebras with the monoid taken to be conjunction, the set of all formal languages over a given alphabet Σ under concatenation, the set of all binary relations on a given set X under relational composition, and more generally the power set of any equivalence relation, again under relational composition. The original application was to relation algebras as a finitely axiomatized generalization of the binary relation example, but there exist interesting examples of residuated Boolean algebras that are not relation algebras, such as the language example. (en) 在数学中，剩余布尔代数是其格结构是布尔代数的剩余格。例子包括幺半群乘法选取为合取的布尔代数，在串接运算之下的给定字母表 Σ 的所有形式语言的集合，在关系复合运算之下的给定集合 X 上所有二元关系的集合，和更一般的在关系复合之下的任何等价类的幂集。最初的应用是作为关系代数中二元关系例子的有限公理化推广，但是存在不是关系代数的有趣的剩余布尔代数的例子，比如语言例子。 (zh) |
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| rdfs:comment | In mathematics, a residuated Boolean algebra is a residuated lattice whose lattice structure is that of a Boolean algebra. Examples include Boolean algebras with the monoid taken to be conjunction, the set of all formal languages over a given alphabet Σ under concatenation, the set of all binary relations on a given set X under relational composition, and more generally the power set of any equivalence relation, again under relational composition. The original application was to relation algebras as a finitely axiomatized generalization of the binary relation example, but there exist interesting examples of residuated Boolean algebras that are not relation algebras, such as the language example. (en) 在数学中，剩余布尔代数是其格结构是布尔代数的剩余格。例子包括幺半群乘法选取为合取的布尔代数，在串接运算之下的给定字母表 Σ 的所有形式语言的集合，在关系复合运算之下的给定集合 X 上所有二元关系的集合，和更一般的在关系复合之下的任何等价类的幂集。最初的应用是作为关系代数中二元关系例子的有限公理化推广，但是存在不是关系代数的有趣的剩余布尔代数的例子，比如语言例子。 (zh) |
| rdfs:label | Residuated Boolean algebra (en) 剩余布尔代数 (zh) |
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