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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. ==Definition== A residuated Boolean algebra is an algebraic structure (''L'', ∧, ∨, ¬, 0, 1, •, I, \, /) such that : (i) (''L'', ∧, ∨, •, I, \, /) is a residuated lattice, and :(ii) (''L'', ∧, ∨, ¬, 0, 1) is a Boolean algebra. An equivalent signature better suited to the relation algebra application is (''L'', ∧, ∨, ¬, 0, 1, •, I, ▷, ◁) where the unary operations ''x''\ and ''x''▷ are intertranslatable in the manner of De Morgan's laws via :''x''\''y'' = ¬(''x''▷¬''y''), ''x''▷''y'' = ¬(''x''\¬''y''), and dually /''y'' and ◁''y'' as : ''x''/''y'' = ¬(¬''x''◁''y''), ''x''◁''y'' = ¬(¬''x''/''y''), with the residuation axioms in the residuated lattice article reorganized accordingly (replacing ''z'' by ¬''z'') to read :(''x''▷''z'')∧''y'' = 0 ⇔ (''x''•''y'')∧''z'' = 0 ⇔ (''z''◁''y'')∧''x'' = 0 This De Morgan dual reformulation is motivated and discussed in more detail in the section below on conjugacy. Since residuated lattices and Boolean algebras are each definable with finitely many equations, so are residuated Boolean algebras, whence they form a finitely axiomatizable variety. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Residuated Boolean algebra」の詳細全文を読む スポンサード リンク
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