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.

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  • 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.
  • 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。
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  • 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.
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  • Residuated Boolean algebra
  • 剩余布尔代数
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