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- Boolean_algebras_canonically_defined abstract "Boolean algebras have been formally defined variously as a kind of lattice and as a kind of ring. This article presents them, equally formally, as simply the models of the equational theory of two values, and observes the equivalence of both the lattice and ring definitions to this more elementary one.Boolean algebra is a mathematically rich branch of abstract algebra. Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the equational theory of the two values 0 and 1 (whose interpretation need not be numerical). Common to Boolean algebras, groups, and vector spaces is the notion of an algebraic structure, a set closed under zero or more operations satisfying certain equations.Just as there are basic examples of groups, such as the group Z of integers and the permutation group Sn of permutations of n objects, there are also basic examples of Boolean algebra such as the following.The algebra of binary digits or bits 0 and 1 under the logical operations including disjunction, conjunction, and negation. Applications include the propositional calculus and the theory of digital circuits.The algebra of sets under the set operations including union, intersection, and complement. Applications include any area of mathematics for which sets form a natural foundation.Boolean algebra thus permits applying the methods of abstract algebra to mathematical logic, digital logic, and the set-theoretic foundations of mathematics.Unlike groups of finite order, which exhibit complexity and diversity and whose first-order theory is decidable only in special cases, all finite Boolean algebras share the same theorems and have a decidable first-order theory. Instead the intricacies of Boolean algebra are divided between the structure of infinite algebras and the algorithmic complexity of their syntactic structure.".
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- Boolean_algebras_canonically_defined wikiPageWikiLink Abstract_algebra.
- Boolean_algebras_canonically_defined wikiPageWikiLink Addition.
- Boolean_algebras_canonically_defined wikiPageWikiLink Affine_transformation.
- Boolean_algebras_canonically_defined wikiPageWikiLink Alan_Turing.
- Boolean_algebras_canonically_defined wikiPageWikiLink Alfred_Tarski.
- Boolean_algebras_canonically_defined wikiPageWikiLink Alfred_W._Hales.
- Boolean_algebras_canonically_defined wikiPageWikiLink Algebra_of_sets.
- Boolean_algebras_canonically_defined wikiPageWikiLink Algebraic_structure.
- Boolean_algebras_canonically_defined wikiPageWikiLink Algorithm.
- Boolean_algebras_canonically_defined wikiPageWikiLink Arity.
- Boolean_algebras_canonically_defined wikiPageWikiLink Associative.
- Boolean_algebras_canonically_defined wikiPageWikiLink Associative_property.
- Boolean_algebras_canonically_defined wikiPageWikiLink Atom_(order_theory).
- Boolean_algebras_canonically_defined wikiPageWikiLink Axiom_schema.
- Boolean_algebras_canonically_defined wikiPageWikiLink Axiomatic_system.
- Boolean_algebras_canonically_defined wikiPageWikiLink Axiomatization.
- Boolean_algebras_canonically_defined wikiPageWikiLink Bilinear.
- Boolean_algebras_canonically_defined wikiPageWikiLink Binary_digit.
- Boolean_algebras_canonically_defined wikiPageWikiLink Binary_operation.
- Boolean_algebras_canonically_defined wikiPageWikiLink Birkhoffs_HSP_theorem.
- Boolean_algebras_canonically_defined wikiPageWikiLink Bit.
- Boolean_algebras_canonically_defined wikiPageWikiLink Bit_array.
- Boolean_algebras_canonically_defined wikiPageWikiLink Bit_vector.
- Boolean_algebras_canonically_defined wikiPageWikiLink Boolean-valued_function.
- Boolean_algebras_canonically_defined wikiPageWikiLink Boolean-valued_model.
- Boolean_algebras_canonically_defined wikiPageWikiLink Boolean_algebra.
- Boolean_algebras_canonically_defined wikiPageWikiLink Boolean_algebra_(structure).
- Boolean_algebras_canonically_defined wikiPageWikiLink Boolean_algebras.
- Boolean_algebras_canonically_defined wikiPageWikiLink Boolean_algebras_canonically_defined.
- Boolean_algebras_canonically_defined wikiPageWikiLink Boolean_domain.
- Boolean_algebras_canonically_defined wikiPageWikiLink Boolean_function.
- Boolean_algebras_canonically_defined wikiPageWikiLink Boolean_logic.
- Boolean_algebras_canonically_defined wikiPageWikiLink Boolean_ring.
- Boolean_algebras_canonically_defined wikiPageWikiLink Cantors_diagonal_argument.
- Boolean_algebras_canonically_defined wikiPageWikiLink Cardinality.
- Boolean_algebras_canonically_defined wikiPageWikiLink Cartesian_closed_category.
- Boolean_algebras_canonically_defined wikiPageWikiLink Category:Articles_with_inconsistent_citation_formats.
- Boolean_algebras_canonically_defined wikiPageWikiLink Category:Boolean_algebra.
- Boolean_algebras_canonically_defined wikiPageWikiLink Category_(mathematics).
- Boolean_algebras_canonically_defined wikiPageWikiLink Charles_Sanders_Peirce.
- Boolean_algebras_canonically_defined wikiPageWikiLink Clopen_set.
- Boolean_algebras_canonically_defined wikiPageWikiLink Closed_monoidal_category.
- Boolean_algebras_canonically_defined wikiPageWikiLink Commutative.
- Boolean_algebras_canonically_defined wikiPageWikiLink Commutative_property.
- Boolean_algebras_canonically_defined wikiPageWikiLink Compact_Hausdorff_space.
- Boolean_algebras_canonically_defined wikiPageWikiLink Compact_space.
- Boolean_algebras_canonically_defined wikiPageWikiLink Complement_(order_theory).
- Boolean_algebras_canonically_defined wikiPageWikiLink Complement_(set_theory).
- Boolean_algebras_canonically_defined wikiPageWikiLink Complemented_lattice.
- Boolean_algebras_canonically_defined wikiPageWikiLink Complete_Boolean_algebra.
- Boolean_algebras_canonically_defined wikiPageWikiLink Countable_set.
- Boolean_algebras_canonically_defined wikiPageWikiLink Countably_infinite.
- Boolean_algebras_canonically_defined wikiPageWikiLink De_Morgans_laws.
- Boolean_algebras_canonically_defined wikiPageWikiLink Decidability_(logic).
- Boolean_algebras_canonically_defined wikiPageWikiLink Digital_circuit.
- Boolean_algebras_canonically_defined wikiPageWikiLink Digital_circuits.
- Boolean_algebras_canonically_defined wikiPageWikiLink Digital_electronics.
- Boolean_algebras_canonically_defined wikiPageWikiLink Digital_logic.
- Boolean_algebras_canonically_defined wikiPageWikiLink Direct_product.
- Boolean_algebras_canonically_defined wikiPageWikiLink Discrete_space.
- Boolean_algebras_canonically_defined wikiPageWikiLink Distributive_lattice.
- Boolean_algebras_canonically_defined wikiPageWikiLink Emil_Leon_Post.
- Boolean_algebras_canonically_defined wikiPageWikiLink Emil_Post.
- Boolean_algebras_canonically_defined wikiPageWikiLink Equational_theory.
- Boolean_algebras_canonically_defined wikiPageWikiLink Field_of_sets.
- Boolean_algebras_canonically_defined wikiPageWikiLink Filter_(mathematics).
- Boolean_algebras_canonically_defined wikiPageWikiLink Filtered_category.
- Boolean_algebras_canonically_defined wikiPageWikiLink Filtered_colimit.
- Boolean_algebras_canonically_defined wikiPageWikiLink Finitary.
- Boolean_algebras_canonically_defined wikiPageWikiLink Finitary_boolean_function.
- Boolean_algebras_canonically_defined wikiPageWikiLink First-order_logic.
- Boolean_algebras_canonically_defined wikiPageWikiLink Foundation_of_mathematics.
- Boolean_algebras_canonically_defined wikiPageWikiLink Foundations_of_mathematics.
- Boolean_algebras_canonically_defined wikiPageWikiLink Free_Boolean_algebra.
- Boolean_algebras_canonically_defined wikiPageWikiLink Free_object.
- Boolean_algebras_canonically_defined wikiPageWikiLink Functional_completeness.
- Boolean_algebras_canonically_defined wikiPageWikiLink Gate_array.
- Boolean_algebras_canonically_defined wikiPageWikiLink Generating_set_of_an_algebra.
- Boolean_algebras_canonically_defined wikiPageWikiLink Georg_Cantor.
- Boolean_algebras_canonically_defined wikiPageWikiLink George_Boole.
- Boolean_algebras_canonically_defined wikiPageWikiLink Group_(mathematics).
- Boolean_algebras_canonically_defined wikiPageWikiLink Group_theory.
- Boolean_algebras_canonically_defined wikiPageWikiLink Gödel_number.
- Boolean_algebras_canonically_defined wikiPageWikiLink Gödel_numbering.
- Boolean_algebras_canonically_defined wikiPageWikiLink Hausdorff_space.
- Boolean_algebras_canonically_defined wikiPageWikiLink Homomorphism.
- Boolean_algebras_canonically_defined wikiPageWikiLink Ideal_(order_theory).
- Boolean_algebras_canonically_defined wikiPageWikiLink If_and_only_if.
- Boolean_algebras_canonically_defined wikiPageWikiLink Iff.
- Boolean_algebras_canonically_defined wikiPageWikiLink Index_set.
- Boolean_algebras_canonically_defined wikiPageWikiLink Indexed_family.
- Boolean_algebras_canonically_defined wikiPageWikiLink Infimum.
- Boolean_algebras_canonically_defined wikiPageWikiLink Infimum_and_supremum.
- Boolean_algebras_canonically_defined wikiPageWikiLink Integer.