Moufangs

Moufangs, commonly known as Moufang loops, are a class of algebraic structures consisting of a set with a binary operation and a distinguished identity element, in which every element has a two-sided inverse but the operation need not be associative. They are named after Ruth Moufang, who studied them in the 1930s. The defining Moufang identities can be presented in several equivalent forms; the most standard is (xy)(zx) = x(yz)x for all elements x, y, z in the loop. Equivalently, (ab)(ca) = a(bc)a also characterizes the same class.

A key property of Moufang loops is diassociativity: the subloop generated by any two elements is associative, hence forms a group. This implies, in particular, that two elements determine a predictable, group-like structure within the broader nonassociative system. Moufang loops are also flexible, satisfying (xy)x = x(yx) for all elements x and y.

Examples and constructions: The set of nonzero elements of the octonions under multiplication forms a Moufang loop. More generally, the multiplicative loop of any alternative division algebra is Moufang; the unit group of a composition algebra is likewise Moufang. These examples connect Moufang loops to broader themes in nonassociative algebra and division algebras.

Connections and applications: Moufang loops appear in projective geometry (notably Moufang planes), in the theory of triality in Lie theory, and in the structural study of nonassociative algebras. They provide important natural examples of nonassociative systems with strong symmetry and diassociativity properties.

History: The concept is named for Ruth Moufang, whose work linked these loops to octonions and geometric constructions, contributing to the development of nonassociative algebra in the 20th century.