Flexible algebra
In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity:
for any two elements a and b of the set. A magma is flexible if the binary operation with which it is equipped is flexible. Similarly, a nonassociative algebra is flexible if its multiplication operator is flexible.
Every commutative or associative operation is flexible, so flexibility becomes important for binary operations that are neither commutative nor associative, e.g. for the multiplication of sedenions, which are not even alternative.
In 1954, Richard Schafer examined the algebras generated by the Cayley–Dickson process over a field and showed that they satisfy the flexible identity.Examples
Besides associative algebras, the following classes of nonassociative algebras are flexible:
Similarly, the following classes of nonassociative magmas are flexible:
The sedenions, and all algebras constructed from these by iterating the Cayley–Dickson construction, are also flexible.
