# Alternativity

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{{Short description|Property of a binary operation}}
{{distinguish|Alternatization}}
{{Technical|date=November 2021}}
{{one source |date=May 2024}}
In [abstract algebra](/source/abstract_algebra), '''alternativity''' is a property of a [binary operation](/source/binary_operation).  A [magma](/source/Magma_(algebra)) {{mvar|G}} is said to be '''{{visible anchor|left alternative}}''' if <math>(xx)y = x(xy)</math> for all <math>x, y \in G</math> and '''{{visible anchor|right alternative}}''' if <math>y(xx) = (yx)x</math> for all <math>x, y \in G</math>. A magma that is both left and right alternative is said to be '''{{visible anchor|alternative}}'''.<ref>{{citation
 | last1 = Phillips | first1 = J. D.
 | last2 = Stanovský | first2 = David
 | doi = 10.3233/AIC-2010-0460 
 | journal = AI Communications
 | mr = 2647941 | zbl=1204.68181
 | pages = 267–283
 | title = Automated theorem proving in quasigroup and loop theory
 | url = http://www.karlin.mff.cuni.cz/~stanovsk/math/qptp.pdf
 | volume = 23 | issue=2–3
 | year = 2010}}.</ref>

Any [associative](/source/Associativity) magma (that is, a [semigroup](/source/semigroup)) is alternative. More generally, a magma in which every pair of elements generates an associative submagma must be alternative. The [converse](/source/converse_(logic)), however, is not true, in contrast to the situation in [alternative algebra](/source/alternative_algebra)s.

==Examples==
Examples of algebraic structures with an alternative multiplication include:
* Any [semigroup](/source/semigroup) is associative and therefore alternative.
* [Moufang loop](/source/Moufang_loop)s are alternative and flexible but generally not associative. See {{Section link|Moufang loop|Examples}} for more examples.
* [Octonion](/source/Octonion) multiplication is alternative and flexible.  The same is more generally true for any [octonion algebra](/source/octonion_algebra).
* Applying the [Cayley-Dickson construction](/source/Cayley-Dickson_construction) once to a [commutative ring](/source/commutative_ring) with a trivial involution <math>a^\ast = a</math> gives a commutative associative algebra.  Applying it twice gives an associative algebra.  Applying it three times gives an alternative algebra.  Applying it four or more times gives an algebra that is typically not alternative (thought it is in characteristic two).  An example is the sequence <math>\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}, \mathbb{S},...</math> where <math>\mathbb{H}</math> is the algebra of quaternions, <math>\mathbb{O}</math> is the algebra of [octonions](/source/octonions), and <math>\mathbb{S}</math> is the algebras of [sedenions](/source/sedenions).

==See also==
* [Flexible algebra](/source/Flexible_algebra)
* [Power associativity](/source/Power_associativity)

==References==
{{reflist}}

* {{ cite book | last=Schafer  | first=Richard D. |author-link=Richard D. Schafer| year=1995 | orig-year=1966 | title=An Introduction to Nonassociative Algebras | publisher=Dover | isbn=0-486-68813-5 | url=https://books.google.com/books?isbn=0486688135 | zbl=0145.25601 }}

Category:Properties of binary operations

{{algebra-stub}}

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