Monadic Boolean algebra (original) (raw)
Monadyczna algebra Boole’a – algebra Boole’a z dodatkowym działaniem jednoargumentowym które spełnia pewne warunki naśladujące własności kwantyfikatora egzystencjalnego.
| Property | Value |
|---|---|
| dbo:abstract | In abstract algebra, a monadic Boolean algebra is an algebraic structure A with signature ⟨·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩, where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra. The monadic/unary operator ∃ denotes the existential quantifier, which satisfies the identities (using the received prefix notation for ∃): * ∃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 definition and notation that take ∀ as primitive and ∃ as defined, so that ∃x := (∀x ' )' . (Compare this with the definition of the dual Boolean algebra.) Hence, with this notation, an algebra A has signature ⟨·, +, ', 0, 1, ∀⟩, with ⟨A, ·, +, ', 0, 1⟩ a Boolean algebra, as before. Moreover, ∀ satisfies the following dualized version of the above identities: 1. * ∀1 = 1 2. * ∀x ≤ x 3. * ∀(xy) = ∀x∀y 4. * ∀x + ∀y = ∀(x + ∀y). ∀x is the universal closure of x. (en) Monadyczna algebra Boole’a – algebra Boole’a z dodatkowym działaniem jednoargumentowym które spełnia pewne warunki naśladujące własności kwantyfikatora egzystencjalnego. (pl) 在抽象代数中，一元布尔代数是带有如下（signature）的代数结构 <A, ·, +, ', 0, 1, ∃> 有型 <2,2,1,0,0,1>， 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： 1. * ∃0 = 0 2. * ∃x ≥ x 3. * ∃(x + y) = ∃x + ∃y 4. * ∃x∃y = ∃(x∃y). ∃x 是 x 的“存在闭包”。于 ∃ 的是一元算子 ∀，它是全称量词，定义为 ∀x := (∃x' )'。 一元布尔代数有公式，取 ∀ 为原始，把 ∃ 定义为 ∃x := (∀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 = 1 2. * ∀x ≤ x 3. * ∀(xy) = ∀x∀y 4. * ∀x + ∀y = ∀(x + ∀y). ∀x 是 x 的“全称闭包”。 (zh) |
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| rdfs:comment | Monadyczna algebra Boole’a – algebra Boole’a z dodatkowym działaniem jednoargumentowym które spełnia pewne warunki naśladujące własności kwantyfikatora egzystencjalnego. (pl) 在抽象代数中，一元布尔代数是带有如下（signature）的代数结构 <A, ·, +, ', 0, 1, ∃> 有型 <2,2,1,0,0,1>， 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： 1. * ∃0 = 0 2. * ∃x ≥ x 3. * ∃(x + y) = ∃x + ∃y 4. * ∃x∃y = ∃(x∃y). ∃x 是 x 的“存在闭包”。于 ∃ 的是一元算子 ∀，它是全称量词，定义为 ∀x := (∃x' )'。 一元布尔代数有公式，取 ∀ 为原始，把 ∃ 定义为 ∃x := (∀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 = 1 2. * ∀x ≤ x 3. * ∀(xy) = ∀x∀y 4. * ∀x + ∀y = ∀(x + ∀y). ∀x 是 x 的“全称闭包”。 (zh) In abstract algebra, a monadic Boolean algebra is an algebraic structure A with signature ⟨·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩, where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra. The monadic/unary operator ∃ denotes the existential quantifier, which satisfies the identities (using the received prefix notation for ∃): * ∃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' )'. 1. * ∀1 = 1 2. * ∀x ≤ x 3. * ∀(xy) = ∀x∀y 4. * ∀x + ∀y = ∀(x + ∀y). (en) |
| rdfs:label | Monadic Boolean algebra (en) Monadyczna algebra Boole’a (pl) 一元布尔代数 (zh) |
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