Flexible algebra

In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity:

${\displaystyle a\bullet \left(b\bullet a\right)=\left(a\bullet b\right)\bullet a}$

for any two elements a and b of the set. A magma (that is a set equipped with a binary operation) is flexible if the binary operation that it is equipped with 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.[1]

Examples

Besides associative algebras, the following classes of nonassociative algebras are flexible:

Similarly, the following classes of nonassociative magmas are flexible: