Editing Alternativity

{{Short description|Property of a binary operation}}
{{distinguish|Alternatization}}
{{Technical|date=November 2021}}
{{one source |date=May 2024}}
In [[abstract algebra]], '''alternativity''' is a property of a [[binary operation]].  A [[Magma (algebra)|magma]] {{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}}''' ('''{{visible anchor|flexible}}''').<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 [[Associativity|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 (logic)|converse]], however, is not true, in contrast to the situation in [[alternative algebra]]s.

==Examples==
Examples of alternative algebras include:
* Any [[Semigroup]] is associative and therefore alternative.
* [[Moufang loop]]s are alternative and flexible but not associative. See {{Section link|Moufang loop|Examples}} for more examples.
* [[Octonion]] multiplication is alternative and flexible.
** More generally [[Cayley-Dickson construction|Cayley-Dickson algebra]] over a [[commutative ring]] is alternative.

==See also==
* [[Flexible algebra]]
* [[Power associativity]]

==References==
{{reflist}}

[[Category:Properties of binary operations]]


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