In abstract algebra, a monadic Boolean algebra is an algebraic structure with signature ⟨A, ·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩, where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra. The prefixed unary operator ∃ denotes the existential quantifier, which satisfies the identities: ∃0 = 0 ∃x ≥ x ∃(x + y) = ∃x + ∃y ∃x∃y = ∃(x∃y). ∃x is the existential closure of x.

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  • In abstract algebra, a monadic Boolean algebra is an algebraic structure with signature ⟨A, ·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩, where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra. The prefixed unary operator ∃ denotes the existential quantifier, which satisfies the identities: ∃0 = 0 ∃x ≥ x ∃(x + y) = ∃x + ∃y ∃x∃y = ∃(x∃y). ∃x is the existential closure of x. Dual to ∃ is the unary operator ∀, the universal quantifier, defined as ∀x := (∃x')'. A monadic Boolean algebra has a dual formulation that takes ∀ as primitive and ∃ as defined, so that ∃x := ' . Hence the dual algebra has signature ⟨A, ·, +, ', 0, 1, ∀⟩, with ⟨A, ·, +, ', 0, 1⟩ a Boolean algebra, as before. Moreover, ∀ satisfies the following dualized version of the above identities: ∀1 = 1 ∀x ≤ x ∀(xy) = ∀x∀y ∀x + ∀y = ∀(x + ∀y). ∀x is the universal closure of x.
  • Monadyczna algebra Boole'a – algebra Boole'a z dodatkowym działaniem jednoargumentowym <math>\exists</math>, które spełnia pewne warunki naśladujące własności kwantyfikatora egzystencjalnego.
  • 在抽象代数中，一元布尔代数是带有如下标识（signature）的代数结构 这里的 <A, ·, +, ', 0, 1> 是布尔代数。 前缀一元算子 ∃ 指示存在量词，它满足恒等式： ∃0 = 0 ∃x ≥ x ∃(x + y) = ∃x + ∃y ∃x∃y = ∃(x∃y). ∃x 是 x 的“存在闭包”。对偶于 ∃ 的是一元算子 ∀，它是全称量词，定义为 ∀x := (∃x')'。 一元布尔代数有对偶公式，取 ∀ 为原始，把 ∃ 定义为 ∃x := ' 。所以对偶的代数有标识 <A, ·, +, ', 0, 1, ∀>，带有 <A, ·, +, ', 0, 1> 是布尔代数。此外，∀ 满足上面恒等式的对偶版本： ∀1 = 1 ∀x ≤ x ∀(xy) = ∀x∀y ∀x + ∀y = ∀(x + ∀y). ∀x 是 x 的“全称闭包”。
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  • In abstract algebra, a monadic Boolean algebra is an algebraic structure with signature ⟨A, ·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩, where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra. The prefixed unary operator ∃ denotes the existential quantifier, which satisfies the identities: ∃0 = 0 ∃x ≥ x ∃(x + y) = ∃x + ∃y ∃x∃y = ∃(x∃y). ∃x is the existential closure of x.
  • Monadyczna algebra Boole'a – algebra Boole'a z dodatkowym działaniem jednoargumentowym <math>\exists</math>, które spełnia pewne warunki naśladujące własności kwantyfikatora egzystencjalnego.
  • 在抽象代数中，一元布尔代数是带有如下标识（signature）的代数结构 这里的 <A, ·, +, ', 0, 1> 是布尔代数。 前缀一元算子 ∃ 指示存在量词，它满足恒等式： ∃0 = 0 ∃x ≥ x ∃(x + y) = ∃x + ∃y ∃x∃y = ∃(x∃y).
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  • Monadic Boolean algebra
  • Monadyczna algebra Boole'a
  • 一元布尔代数
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