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 {{mvar|G}} is said to be {{visible anchor|left alternative}} if $(xx)y\; =\; x(xy)$ for all $x,\; y\; \backslash in\; G$ and {{visible anchor|right alternative}} if $y(xx)\; =\; (yx)x$ for all $x,\; y\; \backslash in\; G$. A magma that is both left and right alternative is said to be {{visible anchor|alternative}} ({{visible anchor|flexible}}).{{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}}.

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.