Moufang

Moufang is a surname associated with Ruth Moufang (1905–1977), a German mathematician whose work in nonassociative algebra and incidence geometry led to several concepts that bear her name. In mathematics, the term Moufang is attached to structures that satisfy the Moufang condition, a particular kind of identity that constrains nonassociative operations.

A Moufang loop is a loop—an algebraic structure with a binary operation, an identity element, and a division property—that satisfies the Moufang identity. One common form is (xy)(zx) = x[(y z) x] for all elements x, y, z in the loop. There are several equivalent formulations of the Moufang identities. Moufang loops are diassociative, meaning any two elements generate an associative subloop. They generalize groups and include notable examples, such as the unit octonions under multiplication, which form a nonassociative Moufang loop.

In geometry, a Moufang plane is a projective plane that can be coordinatized by a division algebra satisfying the Moufang condition, most famously the octonions. The octonion projective plane, also known as the Cayley plane, is a non-Desarguesian Moufang plane and serves as a key example linking nonassociative algebra with incidence geometry.

The Moufang concept has influenced broader areas of algebra and geometry, including connections to exceptional groups and related symmetry structures. Ruth Moufang’s work thus established a lasting bridge between nonassociative algebraic identities and geometric constructions, with the term Moufang now applied to both algebraic loops and geometric planes named in her honor.