About: Monadic Boolean algebra

Facets (new session)
Description
Metadata
Settings
- Rule:
- Inverse Functional Properties:
- "Same As":

About: Monadic Boolean algebra Goto Sponge NotDistinct Permalink

An Entity of Type : dbo:Building, within Data Space : dbpedia.org associated with source document(s)
QRcode icon

http://dbpedia.org/describe/?url=http%3A%2F%2Fdbpedia.org%2Fresource%2FMonadic_Boolean_algebra

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).

Attributes	Values
rdf:type	yago:WikicatClosureOperators yago:Abstraction100002137 yago:Function113783816 yago:MathematicalRelation113783581 yago:Operator113786413 yago:Relation100031921 building
rdfs:label	Monadic Boolean algebra (en) Monadyczna algebra Boole’a (pl) 一元布尔代数 (zh)
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)
dct:subject	Boolean algebra Algebraic logic Closure operators
Wikipage page ID	1018197 (xsd:integer)
Wikipage revision ID	1014607548 (xsd:integer)
Link from a Wikipage to another Wikipage	Modal logic Łukasiewicz–Moisil algebra Algebraic structure Paul Halmos Cylindric algebra Interior algebra Prefix Universal quantifier Lindenbaum–Tarski algebra Closure operator Clopen set Propositional logic Synonym Topology Duality (order theory) First-order logic Boolean algebra Semisimple algebra Abstract algebra Algebraic logic Closure operators Boolean algebra (structure) Kuratowski closure axioms Signature (logic) Variety (universal algebra) Polyadic algebra Interior operator Closure algebra Monadic (arity) Monadic logic Existential quantifier Unary operator
sameAs	Monadic Boolean algebra Monadic Boolean algebra Monadic Boolean algebra Monadic Boolean algebra Monadic Boolean algebra Monadic Boolean algebra
dbp:wikiPageUsesTemplate	dbt:Logic-stub
has 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)
gold:hypernym	Structure
prov:wasDerivedFrom	wikipedia-en:Monadic_Boolean_algebra?oldid=1014607548&ns=0
page length (characters) of wiki page	3749 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Monadic_Boolean_algebra
is Link from a Wikipage to another Wikipage of	Łukasiewicz–Moisil algebra Paul Halmos Cylindric algebra Interior algebra List of Boolean algebra topics Algebraic logic Outline of logic S5 algebra
is Wikipage redirect of	S5 algebra
is foaf:primaryTopic of	wikipedia-en:Monadic_Boolean_algebra

Faceted Search & Find service v1.17_git145 as of Aug 30 2024

Alternative Linked Data Documents: ODE Content Formats:

RDF

ODATA

Microdata

About

OpenLink Virtuoso version 08.03.3331 as of Sep 2 2024, on Linux (x86_64-generic-linux-glibc212), Single-Server Edition (378 GB total memory, 52 GB memory in use)
Data on this page belongs to its respective rights holders.
Virtuoso Faceted Browser Copyright © 2009-2024 OpenLink Software