Alternativity
Property of a binary operation
In abstract algebra, alternativity is a property of a binary operation. A magma G is said to be left alternative if 
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.
01Examples
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. The same is more generally true for any octonion algebra.
- Applying the Cayley-Dickson construction once to a commutative ring with a trivial involution
gives a commutative associative algebra. Applying it twice gives an associative algebra. Applying it three times gives an alternative algebra. Applying it four or more times gives an algebra that is typically not alternative (counterexamples can be found with characteristic two). An example is the sequence
where
is the algebra of quaternions,
is the algebra of octonions, and
is the algebras of sedenions.
Sources and credits
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