In abstract algebra , alternativity is a property of a binary operation . A magma algebra magma G is said to be left alternative if xx y x xy for all x and y in G and right alternative if y xx yx x for all x and y in G . A magma that is both left and right alternative is said to be alternative . Any associativity associative magma semigroup is clearly 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 algebra s. In fact, an alternative magma need not even be power associative . Category Abstract algebra Category Binary operations Alternativity algebra stub de Alternativit t fr Alternativit nl Alternativiteit pl Alternatywno uk ... more details
Summary Non free media rationale Non free image data Description Culture Crash vol. 1, 14 2004 . Alternate cover for the Special C3 Cosplayer Spotlight Section. Photography by Gari Buenavista. Source Scan made by the original uploader, user Alternativity Portion Full alternate front cover as published Low resolution 330px size unsuitable for high end reproduction other information Non free image rationale Article Culture Crash Comics Purpose Illustration of a specific point within the article. Specifically, the closing of the publication. This unique cover is of what would be, unknown to the publisher at publication time, the last issue of Culture Crash Comics , and also features the first magazine appearance of Alodia Gosiengfiao , at the point where she and her sister Ashley first came to the awareness of the general Philippine public. Replaceability No free use image available. This unique issue is of a copyrighted publication, a freely licensed alternative could not reasonably be obtained. The 330px size is already unsuitable for high end reproduction, but the uploader is willing to further reduce that if requested. However, other lowres image files on wikipedia are of a higher resolution than that, which served as the basis for setting the resolution. Non free image rationale Article Alodia Gosiengfiao Purpose Illustration of a specific point within the article. Specifically, Gosiengfiao s first major media appearance. This unique cover is of what would be, unknown to the publisher at publication time, the last issue of Culture Crash Comics , and also features the first magazine appearance of Alodia Gosiengfiao , at the point where she and her sister Ashley first came to the awareness of the general Philippine public. Replaceability No free use image available. This unique issue is of a copyrighted publication, a freely licensed alternative could not reasonably be obtained. The 330px size is already unsuitable for high end reproduction, but the uploader is will ... more details
In abstract algebra , power associativity is a property of a binary operation which is a weak form of associativity . 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 multiplied by itself several times, it doesn t matter in which order the multiplications are carried out, so for instance x x xx x xx x xx xx . This is stronger than merely saying that xx x x xx for every x in the algebra, but weaker than alternativity or associativity, which requires that xy z x yz for every x , y , and z in the algebra. Every associative algebra is obviously power associative, but so are all other alternative algebra s like the octonion s, which are non associative and even some non alternative algebras like the sedenion s. Any algebra whose elements are idempotent is also power associative. Exponentiation to the power of any natural number other than 0 number zero can be defined consistently whenever multiplication is power associative. For example, there is no ambiguity as to whether x sup 3 sup should be defined as xx x or as x xx , since these are equal. Exponentiation to the power of zero can also be defined if the operation has an identity element , so the existence of identity elements becomes especially useful in power associative contexts. A nice substitution law holds for real power associative algebras with unit, which basically asserts that multiplication of polynomials works as expected. For f a real polynomial in x , and for any a in such an algebra define f a to be the element of the algebra resulting from the obvious substitution of a into f . Then for any two such polynomials f and g , we have that fg a f a g a . References Citation last1 Albert first1 A. Adrian title Power associative rings jstor 1990399 mr 0027750 year 1948 journal Transactions of the American Mathematical Society issn 0002 9947 volume 64 p ... more details
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 math x xy xx y math math yx x y xx math for all x and y in the algebra. Every associative algebra is obviously alternative, but so too are some strictly Algebra over a field Non associative algebras nonassociative algebras such as the octonion s. The sedenion s, on the other hand, are not alternative. The associator Alternative algebras are so named because they are precisely the algebras for which the associator is alternating form alternating . The associator is a multilinear map trilinear map given by math x,y,z xy z x yz math By definition a multilinear map is alternating if it vanishes whenever two of it arguments are equal. The left and right alternative identities for an algebra are equivalent to math x,x,y 0 math math y,x,x 0. math Both of these identities together imply that the associator is totally skew symmetric . That is, math x sigma 1 , x sigma 2 , x sigma 3 sgn sigma x 1,x 2,x 3 math for any permutation . It follows that math x,y,x 0 math for all x and y . This is equivalent to the so called flexible identity math xy x x yx . math The associator of an alternative algebra is therefore alternating. Conversely, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of left alternative identity math x xy xx y math right alternative identity math yx x y xx math flexible identity math xy x x yx . math is alternative and therefore satisfies all three identities. An alternating associator is always totally skew symmetric. The converse holds so long as the characteristic algebra characteristic of the base field is not 2. Properties Artin s theorem states that in an alternative algebra the subalgebra generated by any two elements is associative . Conversely, any algebra for which this is true is clearly alternative. ... more details
, the Moufang identities imply x xy xx y span style padding 1em span alternativity left alternative identity xy y x yy span style padding 1em span alternativity right alternative identity ... more details
Encyclopedia
top
