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Alternative Algebra
In abstract algebra, an alternative algebra is an algebra over a field, algebra in which multiplication need not be associative, only alternativity, alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. The associator Alternative algebras are so named because they are the algebras for which the associator is alternating form, alternating. The associator is a trilinear map given by :[x,y,z] = (xy)z - x(yz). By definition, a multilinear map is alternating if it Vanish_(mathematics), vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to :[x,x,y] = 0 :[y,x,x] = 0 Both of these identities together imply that: :[x,y,x]=[x,x,x]+[x,y,x]+ :-[x,x+y,x+y] = := [x,x+y,-y] = := [x,x,-y] - [x,y,y] = 0 for all x and y. This is equivalent to the ''f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigintaduonion
In abstract algebra, the trigintaduonions, also known as the , , form a commutative property, noncommutative and associative property, nonassociative algebra over a field, algebra over the real numbers. Names The word ''trigintaduonion'' is derived from Latin ' 'thirty' + ' 'two' + the suffix -''nion'', which is used for hypercomplex number systems. Other names include , , , and . Definition Every trigintaduonion is a linear combination of the unit trigintaduonions e_0, e_1, e_2, e_3, ..., e_, which form a Basis (linear algebra), basis of the vector space of trigintaduonions. Every trigintaduonion can be represented in the form :x = x_0 e_0 + x_1 e_1 + x_2 e_2 + \cdots + x_ e_ + x_ e_ with real coefficients . The trigintaduonions can be obtained by applying the Cayley–Dickson construction to the sedenions. Applying the Cayley–Dickson construction to the trigintaduonions yields a 64-dimensional algebra called the ''64-ions'', ''64-nions'', ''sexagintaquatronions'', or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed Division Algebra
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a nondegenerate positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions, and that there are no other possibilities. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras. The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields. Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwitz problem, solved also i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Ring
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defined abo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Of Units
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this property and is called the multiplicative inverse of . The set of units of forms a group under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ). Less commonly, the term ''unit'' is sometimes used to refer to the element of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also unit matrix. Because of this ambiguity, is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng. Examples The multiplicative identity and its additive inverse are always units. More ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moufang Loop
In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by . Smooth Moufang loops have an associated algebra, the Malcev algebra, similar in some ways to how a Lie group has an associated Lie algebra. Definition A Moufang loop is a loop Q that satisfies the four following equivalent identities for all x, y, z in Q (the binary operation in Q is denoted by juxtaposition): #z(x(zy)) = ((zx)z)y #x(z(yz)) = ((xz)y)z #(zx)(yz) = (z(xy))z #(zx)(yz) = z((xy)z) These identities are known as Moufang identities. Examples * Any group is an associative loop and therefore a Moufang loop. * The nonzero octonions form a nonassociative Moufang loop under octonion multiplication. * The subset of unit norm octonions (forming a 7-sphere in O) is closed under multiplication and therefore forms a Moufang loop. * The subset of unit norm integral octonions is a finite Moufang loop of o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is a right inverse of . (An identity element is an element such that and for all and for which the left-hand sides are defined.) When the operation is associative, if an element has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the ''inverse element'' or simply the ''inverse''. Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse. In a ring, an ''invertible element'', also called a unit, is an element that is invertible under multiplication (this is not ambiguous, as every element is invertible under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moufang Identities
In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by . Smooth Moufang loops have an associated algebra, the Malcev algebra, similar in some ways to how a Lie group has an associated Lie algebra. Definition A Moufang loop is a loop Q that satisfies the four following equivalent identities for all x, y, z in Q (the binary operation in Q is denoted by juxtaposition): #z(x(zy)) = ((zx)z)y #x(z(yz)) = ((xz)y)z #(zx)(yz) = (z(xy))z #(zx)(yz) = z((xy)z) These identities are known as Moufang identities. Examples * Any group is an associative loop and therefore a Moufang loop. * The nonzero octonions form a nonassociative Moufang loop under octonion multiplication. * The subset of unit norm octonions (forming a 7-sphere in O) is closed under multiplication and therefore forms a Moufang loop. * The subset of unit norm integral octonions is a finite Moufang loop of o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power-associative
In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. Definition An algebra over a field, algebra (or more generally a magma (algebra), magma) is said to be power-associative if the subalgebra generated by any element is associative. Concretely, this means that if an element x is performed an operation * by itself several times, it doesn't matter in which order the operations are carried out, so for instance x*(x*(x*x)) = (x*(x*x))*x = (x*x)*(x*x). Examples and properties Every associative algebra is power-associative, but so are all other alternative algebras (like the octonions, which are non-associative) and even non-alternative flexible algebras like the sedenions, trigintaduonions, and Okubo algebras. Any algebra whose elements are idempotent is also power-associative. Exponentiation to the power of any positive integer can be defined consistently whenever multiplication is power-as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another proposition; it might also be used more casually to refer to something which naturally or incidentally accompanies something else. Overview In mathematics, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term ''corollary'', rather than ''proposition'' or ''theorem'', is intrinsically subjective. More formally, proposition ''B'' is a corollary of proposition ''A'', if ''B'' can be readily deduced from ''A'' or is self-evident from its proof. In many cases, a corollary corresponds to a special case of a larger theorem, which makes the theorem easier to use and apply, even though its importance is generally considered to be secondary to that of the theorem. In particular, ''B'' is unlikely to be te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subalgebra
In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of ''all'' algebraic structures. "Subalgebra" can refer to either case. Subalgebras for algebras over a ring or field A subalgebra of an algebra over a commutative ring or field is a vector subspace which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebra over the same ring or field. This notion also applies to most specializations, where the multiplication must satisfy additional properties, e.g. to associative algebras or to Lie algebras. Only for unital algebras is there a stronger notion, of unital subalgebra, for which it is also required that the unit of the subalgebra be the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
