Alternativity

Not to be confused with Alternatization.

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 $x,y\in G$ and right alternative if $y(xx)=(yx)x$ for all $x,y\in G$. A magma that is both left and right alternative is said to be alternative (flexible).

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 alternative algebras include:

• Any Semigroup is associative and therefor alternative.
• Moufang loops are alternative and flexible but not associative. See Moufang loop § Examples for more examples.
• Octonion multiplication is alternative and flexible.
  • More generally Cayley-Dickson algebra over a commutative ring is alternative.

See also

• Flexible algebra
• Power associativity