# Alternativity

In abstract algebra, **alternativity** is a property of a binary operation. A 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**.^{[1]}

Any associative magma (i.e., 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. In fact, an alternative magma need not even be power-associative.

## ReferencesEdit

**^**Phillips, J. D.; Stanovský, David (2010), "Automated theorem proving in quasigroup and loop theory" (PDF),*AI Communications*,**23**(2–3): 267–283, doi:10.3233/AIC-2010-0460, MR 2647941, Zbl 1204.68181.

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