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 (that is, a set equipped with a binary operation) 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 D. Schafer examined the algebras generated by the Cayley–Dickson process over a field and showed that they satisfy the flexible identity. Besides associative algebras, the following classes of nonassociative algebras are flexible: Alternative algebras Lie algebras Jordan algebras (which are commutative) Okubo algebras Similarly, the following classes of nonassociative magmas are flexible: Alternative magmas Semigroups (which are associative magmas, and which are also alternative) The sedenions, and all algebras constructed from these by iterating the Cayley–Dickson construction, are also flexible.

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Alternativity

In abstract algebra, alternativity is a property of a binary operation. A magma G is said to be if for all and if for all 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. In fact, an alternative magma need not even be power-associative.

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In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by . Smooth Moufang loops have an associated algebra, the Malcev algebra, similar in some ways to how a Lie group has an associated Lie algebra. A Moufang loop is a loop that satisfies the four following equivalent identities for all , , in (the binary operation in is denoted by juxtaposition): These identities are known as Moufang identities.

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