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A Boolean algebra can be interpreted either as a special kind of ring (a Boolean ring) or a special kind of distributive lattice (a Boolean lattice). Each interpretation is responsible for different distributive laws in the Boolean algebra. Similar structures without distributive laws are near-rings and near-fields instead of rings and division ...
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 ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
All concrete Boolean algebras satisfy the laws (by proof rather than fiat), whence every concrete Boolean algebra is a Boolean algebra according to our definitions. This axiomatic definition of a Boolean algebra as a set and certain operations satisfying certain laws or axioms by fiat is entirely analogous to the abstract definitions of group ...
An alternative way of stating the same fact is that every distributive lattice is a subdirect product of copies of the two-element chain, or that the only subdirectly irreducible member of the class of distributive lattices is the two-element chain. As a corollary, every Boolean lattice has this property as well. [6]
Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.' [1] Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the ...
The Dedekind–MacNeille completion is characterized among completions of S by this property. [14] The Dedekind–MacNeille completion of a Boolean algebra is a complete Boolean algebra; this result is known as the Glivenko–Stone theorem, after Valery Ivanovich Glivenko and Marshall Stone. [15]
A term's definition may require additional properties that are not listed in this table. This Hasse diagram depicts a partially ordered set with four elements: a , b , the maximal element a ∨ {\displaystyle \vee } b equal to the join of a and b , and the minimal element a ∧ {\displaystyle \wedge } b equal to the meet of a and b .