nonassociative

Nonassociative is a term used in mathematics to describe a binary operation that does not satisfy the associative law in general. An operation is associative if (ab)c = a(bc) for all elements a, b, and c of the underlying set. If this equality fails for at least one triple, the operation is nonassociative.

Common, everyday examples illustrate nonassociativity. Subtraction on real numbers is nonassociative: (a − b) − c ≠ a − (b − c) in general. Division likewise exhibits nonassociativity: (a/b)/c ≠ a/(b/c) for many choices of a, b, and c. Exponentiation is another familiar case; depending on how it is grouped, a^(b^c) and (a^b)^c are not equal in general.

In algebra, many important structures are nonassociative. The Lie bracket [x,y] in a Lie algebra is bilinear and antisymmetric, but it is not associative; instead it satisfies the Jacobi identity, which constrains the failure of associativity. Nonassociative algebras also include Jordan algebras and alternative algebras, each with its own defining identities that relax associativity in different ways. A notable, historically significant example is the octonions, a nonassociative normed division algebra discovered by John Graves and Cayley; octonions arise in certain geometric and physical contexts and are constructed via the Cayley-Dickson process, which yields higher-dimensional algebras that typically lose associativity.

Other related concepts describe controlled nonassociativity. The associator (x,y,z) = (xy)z − x(yz) measures how far an operation is from being associative. Structures such as loops and quasigroups generalize groups by dropping associativity, yet retain weaker forms of the inverse or identity properties. Nonassociativity thus occupies a broad and active area of algebra with applications in geometry, topology, and theoretical physics.