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Flexible Algebra
In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity: : a \bullet \left(b \bullet a\right) = \left(a \bullet b\right) \bullet a for any two elements ''a'' and ''b'' of the set. A magma (that is, a set equipped with a binary operation) is flexible if the binary operation with which it is equipped is flexible. Similarly, a nonassociative algebra is flexible if its multiplication operator is flexible. Every commutative or associative operation is flexible, so flexibility becomes important for binary operations that are neither commutative nor associative, e.g. for the multiplication of sedenions, which are not even alternative. In 1954, Richard D. Schafer examined the algebras generated by the Cayley–Dickson process over a field and showed that they satisfy the flexible identity.Richard D. Schafer (1954) “On the algebras formed by the Cayley-Dickson process”, American Journal of Mathematics 76: 43 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books in the public domain. The original published editions may be scarce or historically significant. Dover republishes these books, making them available at a significantly reduced cost. Classic reprints Dover reprints classic works of literature, classical sheet music, and public-domain images from the 18th and 19th centuries. Dover also publishes an extensive collection of mathematical, scientific, and engineering texts. It often targets its reprints at a niche market, such as woodworking. Starting in 2015, the company branched out into graphic novel reprints, overseen by Dover acquisitions editor and former comics writer and editor Drew Ford. Most Dover reprints are photo facsimiles of the originals, retaining the original pagination ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maltsev Algebra
In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that :xy = -yx and satisfies the Malcev identity :(xy)(xz) = ((xy)z)x + ((yz)x)x + ((zx)x)y. They were first defined by Anatoly Maltsev (1955). Malcev algebras play a role in the theory of Moufang loops that generalizes the role of Lie algebras in the theory of groups. Namely, just as the tangent space of the identity element of a Lie group forms a Lie algebra, the tangent space of the identity of a smooth Moufang loop forms a Malcev algebra. Moreover, just as a Lie group can be recovered from its Lie algebra under certain supplementary conditions, a smooth Moufang loop can be recovered from its Malcev algebra if certain supplementary conditions hold. For example, this is true for a connected, simply connected real-analytic Moufang loop. Examples *Any Lie algebra is a Malcev algebra. *Any alternative algebra may be made into a Mal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zorn Ring
In mathematics, a Zorn ring is an alternative ring in which for every non-nilpotent ''x'' there exists an element ''y'' such that ''xy'' is a non-zero idempotent . named them after Max August Zorn, who studied a similar condition in . For associative rings, the definition of Zorn ring can be restated as follows: the Jacobson radical J(''R'') is a nil ideal and every right ideal of ''R'' which is not contained in J(''R'') contains a nonzero idempotent. Replacing "right ideal" with "left ideal" yields an equivalent definition. Left or right Artinian rings, left or right perfect rings, semiprimary rings and von Neumann regular ring In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the eleme ...s are all examples of associative Zorn rings. References * * * * Non-associative algebras Ring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley–Dickson Construction
In mathematics, the Cayley–Dickson construction, sometimes also known as the Cayley–Dickson process or the Cayley–Dickson procedure produces a sequence of algebra over a field, algebras over the field (mathematics), field of real numbers, each with twice the dimension of a vector space, dimension of the previous one. It is named after Arthur Cayley and Leonard Eugene Dickson. The algebras produced by this process are known as Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently applied in mathematical physics. The Cayley–Dickson construction defines a new algebra as a Cartesian product of an algebra with itself, with multiplication defined in a specific way (different from the componentwise operation, componentwise multiplication) and an involution (mathematics), involution known as ''conjugation''. The product of an element and its complex conjugate, conjugate (or sometimes the square ... [...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] |
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] |
Semigroups
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): , or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. As in the case of groups or magmas, the semigroup operation need not be commutative, so is not necessarily equal to ; a well-known example of an operation that is associative but non-commutative is matrix multiplication. If the semigroup operation is commutative, then the semigroup is called a ''commutative semigroup'' o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Magma
In mathematics, there exist magmas that are commutative but not associative. A simple example of such a magma may be derived from the children's game of rock, paper, scissors. Such magmas give rise to non-associative algebras. A magma which is both commutative and associative is a commutative semigroup. Example: rock, paper, scissors In the game of rock paper scissors, let M := \ , standing for the "rock", "paper" and "scissors" gestures respectively, and consider the binary operation \cdot : M \times M \to M derived from the rules of the game as follows: : For all x, y \in M: :* If x \neq y and x beats y in the game, then x \cdot y = y \cdot x = x :* x \cdot x = x I.e. every x is idempotent. : So that for example: :* r \cdot p = p \cdot r = p "paper beats rock"; :* s \cdot s = s "scissors tie with scissors". This results in the Cayley table: : \begin \cdot & r & p & s\\ \hline r & r & p & r\\ p & p & p & s\\ s & r & s & s \end By definition, the ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternative Magma
In abstract algebra, alternativity is a property of a binary operation. A magma is said to be if (xx)y = x(xy) for all x, y \in G and if y(xx) = (yx)x for all x, y \in G. A magma that is both left and right alternative is said to be ().. Any associative magma (that is, a semigroup) is alternative. More generally, a magma in which every pair of elements generates an associative submagma must be alternative. The converse, however, is not true, in contrast to the situation in alternative algebras. Examples Examples of alternative algebras include: * Any Semigroup is associative and therefore alternative. * Moufang loops are alternative and flexible but not associative. See for more examples. * Octonion multiplication is alternative and flexible. ** More generally Cayley-Dickson algebra over a commutative ring is alternative. See also * Flexible algebra * Power associativity In mathematics, specifically in abstract algebra, power associativity is a property of a binary ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
