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In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).

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  • Boolean algebra (structure)
  • Algèbre de Boole (structure)
  • Булева алгебра
  • Álgebra booleana (estrutura)
  • 布尔代数
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  • Em álgebra abstrata, a álgebra booleana ou álgebra reticulada é um reticulado distribuído complementar. Este tipo de estrutura estrutura algébrica captura propriedades essenciais das operações de conjuntos e operações lógicas. A álgebra booleana pode ser vista como uma generalização do conjunto das partes algébrico ou como um campo de conjuntos, ou seus elementos pode ser vistos como valores verdades generalizados. Ele também é um caso especial da álgebra de De Morgan e da álgebra de Kleene.Um anel booleano é, essencialmente, o mesmo que uma álgebra booleana, com o anel multiplicador correspondendo a uma conjunção ∧, e o anel somador correspondendo a uma disjunção exclusiva ou uma diferença simétrica (não disjunção ∨).
  • In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).
  • En mathématiques, une algèbre de Boole, ou parfois anneau de Boole, est une structure algébrique étudiée en particulier en logique mathématique. Une algèbre de Boole peut être définie soit comme une structure ordonnée particulière — c'est un treillis avec plus grand et plus petit élément, dont chacune des deux opérations de borne inférieure et de borne supérieure est distributive par rapport à l'autre, et dont tout élément possède un complément, soit comme un anneau (unitaire) dont tout élément égale son carré.
  • Булевой алгеброй называется непустое множество A с двумя бинарными операциями (аналог конъюнкции), (аналог дизъюнкции), одной унарной операцией (аналог отрицания) и двумя выделенными элементами: 0 (или Ложь) и 1 (или Истина) такими, что для всех a, b и c из множества A верны следующие аксиомы: В нотации · + ¯ Первые три аксиомы означают, что (A, ,
  • 在抽象代数中,布尔代数是捕获了集合运算和逻辑运算二者的根本性质的一个代数结构(就是说一组元素和服从定义的公理的在这些元素上运算)。特别是,它处理集合运算交集、并集、补集;和逻辑运算与、或、非。 例如,逻辑断言陈述a和它的否定¬a不能都同时为真, , 相似于集合论断言子集A和它的补集AC有空交集, 。 因为真值可以在逻辑电路中表示为二进制数或电平,这种相似性同样扩展到它们,所以布尔代数在电子工程和计算机科学中同在数理逻辑中一样有很多实践应用。在电子工程领域专门化了的布尔代数也叫做逻辑代数,在计算机科学领域专门化了布尔代数也叫做布尔逻辑。 布尔代数也叫做布尔格。关联于格(特殊的偏序集合)是在集合包含A ⊆ B和次序 a ≤ b之间的相似所预示的。考虑{x,y,z}的所有子集按照包含排序的格。这个布尔格是偏序集合,在其中{x} ≤ {x,y}。任何两个格的元素,比如p = {x,y}和q = {y,z},都有一个最小上界,这里是{x,y,z},和一个最大下界,这里是{y}。这预示了最小上界(并或上确界)被表示为同逻辑OR一样的符号p∨q;而最大下界(交或下确界)被表示为同逻辑AND一样的符号p∧q。 这种格释义有助于一般化为海廷代数,它是免除要么一个陈述要么它的否定必须为真的限制的布尔代数。海廷代数对应于直觉逻辑,而布尔代数对应于经典逻辑。
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