Alternativity

In abstract algebra, alternativity is a property of a binary operation. A magma G is said to be left alternative if

(

x

x

)

y

=

x

(

x

y

)

{\displaystyle (xx)y=x(xy)}

for all

x

,

y

∈

G

{\displaystyle x,y\in G}

and right alternative if

y

(

x

x

)

=

(

y

x

)

x

{\displaystyle y(xx)=(yx)x}

for all

x

,

y

∈

G

{\displaystyle x,y\in G}

. A magma that is both left and right alternative is said to be alternative.

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.

Examples

Examples of algebraic structures with an alternative multiplication include:

Any semigroup is associative and therefore alternative.

Moufang loops are alternative and flexible but generally not associative. See Moufang loop § Examples for more examples.

Octonion multiplication is alternative and flexible.