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A single axiom that is satisfied only by NAND or NOR must be of the form "something equals ," since otherwise constant functions would satisfy the equation. With up to six NANDs and two variables, none of the possible axiom systems of this kind work even up to 3-value operators. But with 6 NANDS and 3 variables, 296 of the possible axiom systems work up to 3-value operators, and 100 work up to 4-value operators (Wolfram 2002, p. 809).Of the 25 of these that are not trivially equivalent, it then turns out that only the Wolfram axiomand the axiomwhere denotes the NAND operator, are equivalent to the axioms of Boolean algebra (Wolfram 2002, pp. 808-811 and 1174). These candidate axioms were identified by S. Wolfram in 2000, who also proved that there were no smaller candidates...

Conditions arising in the study of the Robbins axiom and its connection with Boolean algebra. Winkler studied Boolean conditions (such as idempotence or existence of a zero) which would make a Robbins algebra become a Boolean algebra. Winkler showed that each of the conditionswhere denotes OR and denotes NOT, known as the first and second Winkler conditions, suffices. A computer proof demonstrated that every Robbins algebra satisfies the second Winkler condition, from which it follows immediately that all Robbins algebras are Boolean.

Consider a Boolean algebra of subsets generated by a set , which is the set of subsets of that can be obtained by means of a finite number of the set operations union, intersection, and complementation. Then each of the elements of is called a Boolean function generated by (Comtet 1974, p. 185). Each Boolean function has a unique representation (up to order) as a union of complete products. It follows that there are inequivalent Boolean functions for a set with cardinality (Comtet 1974, p. 187).In 1938, Shannon proved that a two-valued Boolean algebra (whose members are most commonly denoted 0 and 1, or false and true) can describe the operation of two-valued electrical switching circuits. The following table gives the truth table for the possible Boolean functions of two binary variables.00000000000100001111100011001111010101010011111111010000111110001100111101010101The names and symbols for these functions are given..

A Boolean algebra is a mathematical structure that is similar to a Boolean ring, but that is defined using the meet and join operators instead of the usual addition and multiplication operators. Explicitly, a Boolean algebra is the partial order on subsets defined by inclusion (Skiena 1990, p. 207), i.e., the Boolean algebra of a set is the set of subsets of that can be obtained by means of a finite number of the set operations union (OR), intersection (AND), and complementation (NOT) (Comtet 1974, p. 185). A Boolean algebra also forms a lattice (Skiena 1990, p. 170), and each of the elements of is called a Boolean function. There are Boolean functions in a Boolean algebra of order (Comtet 1974, p. 186).In 1938, Shannon proved that a two-valued Boolean algebra (whose members are most commonly denoted 0 and 1, or false and true) can describe the operation of two-valued electrical switching circuits. In modern times, Boolean..

The conjecture that the equations for a Robbins algebra, commutativity, associativity,and the Robbins axiomwhere denotes NOT and denotes OR, imply those for a Boolean algebra. The conjecture was finally proven using a computer (McCune 1997).

Building on work of Huntington (1933ab), Robbins conjectured that the equations for a Robbins algebra, commutativity, associativity, and the Robbins axiomwhere denotes NOT and denotes OR, imply those for a Boolean algebra. The conjecture was finally proven using a computer (McCune 1997).

An axiom proposed by Huntington (1933) as part of his definition of a Booleanalgebra,(1)where denotes NOT and denotes OR. Taken together, the three axioms consisting of (1), commutativity(2)and associativity(3)are equivalent to the axioms of Boolean algebra.The Huntington operator can be defined in the WolframLanguage by: Huntington := Function[{x, y}, ! (! x \[Or] y) \[Or] ! (! x \[Or] ! y)]That the Huntington axiom is a true statement in Booleanalgebra can be verified by examining its truth table.TTTTFTFTFFFF

Let be an involutive algebra over the field of complex numbers with involution . Then is a right Hilbert algebra if has an inner product satisfying: 1. For all , is bounded on . 2. . 3. The involution is closable. 4. The linear span of products , , is a dense subalgebra of .

Let be an involutive algebra over the field of complex numbers with involution . Then is a left Hilbert algebra if has an inner product satisfying: 1. For all , is bounded on . 2. . 3. The involution is closable. 4. The linear span of products , , is a dense subalgebra of . Left Hilbert algebras are historically known as generalized Hilbert algebras (Takesaki 1970).A basic result in functional analysis says that if the involution map on a left Hilbert algebra is an antilinear isometry with respect to the inner product , then is also a right Hilbert algebra with respect to the involution . The converse also holds.

Exterior algebra is the algebra of the wedge product, also called an alternating algebra or Grassmann algebra. The study of exterior algebra is also called Ausdehnungslehre or extensions calculus. Exterior algebras are graded algebras.In particular, the exterior algebra of a vector space is the direct sum over in the natural numbers of the vector spaces of alternating differential k-forms on that vector space. The product on this algebra is then the wedge product of forms. The exterior algebra for a vector space is constructed by forming monomials , , , etc., where , , , , , and are vectors in and is wedge product. The sums formed from linear combinations of the monomials are the elements of an exterior algebra.The exterior algebra of a vector space can also bedescribed as a quotient vector space,(1)where is the subspace of -tensors generated by transpositions such as and denotes the vector space tensor product. The equivalence class is denoted..

The semigroup algebra , where is a field and a semigroup, is formally defined in the same way as the group algebra . Similarly, a semigroup ring is a variation of the group ring , where the group is replaced by a semigroup . Usually, it is required that have an identity element so that is a unit ring and is a subring of .The group algebra is the set of all formal expressions(1)where for all and for all but finitely many indices so that for sufficiently large (say, ). Hence, we can write the general element as(2)Assigning(3)defines an isomorphism of -algebras between and the polynomial ring .More generally, if is the subsemigroup of generated by the elements , for , the semigroup algebra is isomorphic to the subalgebra of the polynomial ring generated by the monomials

A division algebra, also called a "division ring" or "skew field," is a ring in which every nonzero element has a multiplicative inverse, but multiplication is not necessarily commutative. Every field is therefore also a division algebra. In French, the term "corps non commutatif" is used to mean division algebra, while "corps" alone means field.Explicitly, a division algebra is a set together with two binary operators satisfying the following conditions: 1. Additive associativity: For all , . 2. Additive commutativity: For all , . 3. Additive identity: There exists an element such that for all , . 4. Additive inverse: For every there exists an element such that . 5. Multiplicative associativity: For all , . 6. Multiplicative identity: There exists an element not equal to 0 such that for all , . 7. Multiplicative inverse: For every not equal to 0, there exists such that . 8. Left and right distributivity:..

Relations in the definition of a Steenrod algebra which state that, for ,where denotes function composition and is the floor function.

There are at least two theorems known as Weierstrass's theorem. The first states that the only hypercomplex number systems with commutative multiplication and addition are the algebra with one unit such that and the Gaussian integers.In harmonic analysis, let be any open set, and let , , ..., be a finite or infinite sequence in (possibly with repetitions) that has no accumulation point in . There exists an analytic function on whose zero set is precisely (Krantz 1999, p. 111). This is also sometimes known as the Weierstrass product theorem.

A variety is a class of algebras that is closed under homomorphisms, subalgebras, and direct products. Examples include the variety of groups, the variety of rings, the variety of lattices. The class of fields (viewed as a subclass of the class of rings) is not a variety, because it is not closed under direct products.Some important varieties, such as the variety of distributive lattices, are locally finite, meaning that their finitely generated algebras are finite. Others, such as the variety of all lattices, are not locally finite. In strong varieties, direct sums of locally finite algebras are locally finite.Note that this type of variety arises in universal algebra and really has nothing to do with algebraic varieties, toric varieties, etc.

Universal algebra studies common properties of all algebraic structures, including groups, rings, fields, lattices, etc.A universal algebra is a pair , where and are sets and for each , is an operation on . The algebra is finitary if each of its operations is finitary.A set of function symbols (or operations) of degree is called a signature (or type). Let be a signature. An algebra is defined by a domain (which is called its carrier or universe) and a mapping that relates a function to each -place function symbol from .Let and be two algebras over the same signature , and their carriers are and , respectively. A mapping is called a homomorphism from to if for every and all ,If a homomorphism is surjective, then it is called epimorphism. If is an epimorphism, then is called a homomorphic image of . If the homomorphism is a bijection, then it is called an isomorphism. On the class of all algebras, define a relation by if and only if there is an isomorphism from..

Let be a finite-dimensional power-associative algebra, then is the vector space direct sumwhere , with is the subspace of defined byfor , where is an idempotent.

If is a graded module and there exists a degree-preserving linear map , then is called a graded algebra.Cohomology is a graded algebra. In addition, the grading set is monoid having a compatibility relation such that if is in the grading of the algebra , and is in the grading of the algebra , then is in the grading of the algebra (where and are multiplied in , and and are multiplied in the index monoid). For example, cohomology of a space is a graded algebra over the integers (i.e., a graded ring), since if is an -dimensional cohomology class and is an -dimensional cohomology class, then the cup product is an dimensional cohomology class.The group ring of a group over a ring is a graded -algebra with grading .

A partial algebra is a pair , where for each , there are an ordinal number and a set such that is a function from into . In case , the operation is called a total operation, and otherwise, it is called a partial operation.

Suppose that and are two algebras and is a unital -bimodule. Thenwith the usual matrix-like addition and matrix-like multiplication is an algebra.An algebra is called a triangular algebra if there exist algebras and and an -bimodule such that is (algebraically) isomorphic tounder matrix-like addition and matrix-like multiplication.For example, the algebra of upper triangular matrices over the complex field may be viewed as a triangular algebra when .

A topological partial algebra is a pair , where is a partial algebra and each of the operations is continuous in the product topology. Examples of topological partial algebras include topological groups, topological vector spaces, and topological fields. Specifically, a topological field is an example of a topological partial algebra that is not a topological algebra in the strict sense of the term.

A topological algebra is a pair , where is an algebra and each of the operations is continuous in the product topology. Examples of topological algebras include topological groups, topological vector spaces, and topological rings.

Let be a set of urelements, and let be an enlargement of the superstructure . Let be a finitary algebra with finitely many fundamental operations. Then the extension monad (in ) of is the (generally external) subalgebra of that is given by(1)It can be shown that for any such algebra , we have(2)and several other interesting characterizations hold for extension monads.Here are some results involving extension monads: 1. An algebra is locally finite if and only if . 2. For any algebra, the following are equivalent: is finitely generated, , and is internal. 3. Let and be algebras, with a function from to . Then is a homomorphism if and only if the restriction of to is a homomorphism. 4. For algebras , ..., we have(3)

Let denote an -algebra, so that is a vector space over and(1)(2)where is vector multiplication which is assumed to be bilinear. Now define(3)where . is said to be tame if is a finite union of subspaces of . A two-dimensional 0-associative algebra is tame, but a four-dimensional 4-associative algebra and a three-dimensional 1-associative algebra need not be tame. It is conjectured that a three-dimensional 2-associative algebra is tame, and proven that a three-dimensional 3-associative algebra is tame if it possesses a multiplicative identity element.

For all , , in an alternative algebra ,(1)(2)(3)(Schafer 1996, p. 28).

Let denote an -algebra, so that is a vector space over and(1)(2)Now define(3)where . An Associative -algebra is commutative if for all . Similarly, a ring is commutative if the multiplication operation is commutative, and a Lie algebra is commutative if the commutator is 0 for every and in the Lie algebra.The term "commutative algebra" also refers to the branch of abstract algebra that studies commutative rings. Commutative algebra is important in algebraic geometry.

A nonassociative algebra named after physicistPascual Jordan which satisfies(1)and(2)The latter is equivalent to the so-called Jordan identity(3)(Schafer 1996, p. 4). An associative algebra with associative product can be made into a Jordan algebra by the Jordan product(4)Division by 2 gives the nice identity , but it must be omitted in characteristic .Unlike the case of a Lie algebra, not every Jordan algebra is isomorphic to a subalgebra of some . Jordan algebras which are isomorphic to a subalgebra are called special Jordan algebras, while those that are not are called exceptional Jordan algebras.

In simple terms, let , , and be members of an algebra. Then the algebra is said to be associative if(1)where denotes multiplication. More formally, let denote an -algebra, so that is a vector space over and(2)(3)Then is said to be -associative if there exists an -dimensional subspace of such that(4)for all and . Here, vector multiplication is assumed to be bilinear. An -dimensional -associative algebra is simply said to be "associative."

Let be an infinite set of urelements, and let be an enlargement of . Let be an algebra. Then is hyperfinitely generated provided that it has a hyperfinite subset such that is the smallest internal subalgebra of that contains . ( is a hyperfinite generating set for in this case.)

Let denote an -algebra, so that is a vector space over and(1)(2)Then is said to be alternative if, for all (3)(4)Here, vector multiplication is assumed to be bilinear.The associator is an alternating function, and the subalgebra generated by two elements is associative.

Given a commutative ring , an R-algebra is a Hopf algebra if it has additional structure given by -algebra homomorphisms(1)(comultiplication) and(2)(counit) and an R-modulehomomorphism(3)(antipode) that satisfy the properties 1. Coassociativity:(4)2. Counitarity:(5)3. Antipode property:(6)where is the identity map on , is the multiplication in , and is the -algebra structure map for , also called the unit map. Faà di Bruno's formula can be cast in a framework that is a special case of a Hopf algebra (Figueroa and Gracia-Bondía 2005).Coassociativity means that the above diagram commutes, meaning if the arrows were reversed and were exchanged for , a diagram illustrating the associativity of the multiplication within would be obtained. And since embeds the ground ring into , the counit maps into . The counitarity property is similarly the dual of that satisfied by . With and , an algebra becomes a bialgebra, but it is the..

The Steenrod algebra has to do with the cohomology operations in singular cohomology with integer mod 2 coefficients. For every and there are natural transformations of functors(1)satisfying: 1. for . 2. for all and all pairs . 3. . 4. The maps commute with the coboundary maps in the long exact sequence of a pair. In other words,(2)is a degree transformation of cohomology theories. 5. (Cartan relation)(3)6. (Adem relations) For ,(4)7. where is the cohomology suspension isomorphism. The existence of these cohomology operations endows the cohomology ring with the structure of a module over the Steenrod algebra , defined to be , where is the free module functor that takes any set and sends it to the free module over that set. We think of as being a graded module, where the th gradation is given by . This makes the tensor algebra into a graded algebra over . is the ideal generated by the elements and for . This makes into a graded algebra.By the definition of..

Let be an involutive algebra over the field of complex numbers with involution . Then is a modular Hilbert algebra if has an inner product and a one-parameter group of automorphisms on , , satisfying: 1. . 2. For all , is bounded (hence, continuous) on . 3. The linear span of products , , is a dense subalgebra of . 4. for all , . 5. . 6. . 7. is an entire function of on . 8. For every real number , the set is dense in . The group is called the group of modular automorphisms.Note that the definition of modular Hilbert algebras is closely related to that of generalized Hilbert algebras in that every modular Hilbert algebra is a generalized Hilbert algebra provided that it satisfies one additional condition, namely that the involution is closable as a linear operator on the real pre-Hilbert space . This relationship is due, in part, to the fact that the properties of both structures were at the core of Tomita's original exposition of what is today the heart of Tomita-Takesaki..

There are at least two distinct (though related) notions of the term Hilbert algebrain functional analysis.In some literature, a linear manifold of a (not necessarily separable) Hilbert space is a Hilbert algebra if the following conditions are satisfied: 1. is dense in . 2. is a ring so that, for any , there is defined an element such that , , , and for any complex number . 3. For any , there exists an adjoint element such that , and . 4. For any , there exists a positive number such that for all . 5. For every , there exists a unique bounded linear operator on such that for all . Moreover, if for an element and for all , then . At least one author defines a Hilbert algebra to be a quasi-Hilbertalgebrafor which for all (Dixmier 1981).

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