In abstract algebra, alternativity is a property of a binary operation. A magma is said to be and . A magma that is both left and right alternative is said to be .

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 for more examples.

• Octonion multiplication is alternative and flexible. The same is more generally true for any octonion algebra.

• Applying the Cayley-Dickson construction once to a commutative ring is the algebra of octonions is the algebras of sedenions.

See also

• Flexible algebra

• Power associativity

References