nLab
Moufang loop

Contents

Idea

A Moufang loop is a set with a binary operation that is similar to that of a group, but does not require associativity. Making this precise is a little fiddly.

Definition

A Moufang loop is a set QQ with a binary operation :Q×QQ\cdot\colon Q\times Q \to Q (a magma) with two-sided unit ee such that left and right multiplication, y:QQy\cdot -\colon Q \to Q and x:QQ-\cdot x\colon Q \to Q respectively, are isomorphisms (i.e. it is a unital quasigroup or loop) and such that the Moufang identities hold:

One may consider the weaker analogous structure without unit ee, the Moufang quasigroup, but the Moufang identities, by a result of Kenneth Kunen imply that the quasigroup is a loop.

Properties

Every element in a Moufang loop has a multiplicative inverse; a priori there are only left and right inverses, but these coincide by the Moufang identities.

Since right and left multiplication give isomorphisms of the underlying set, one can ‘divide’ by any element of the Moufang loop, on the right or on the left (recall we are not assuming commutativity). Thus one can define a Moufang loop as a set together with a multiplication as above, together with right and left division operations /,\:Q×QQ/, \backslash \colon Q\times Q \to Q, again satisfying the Moufang identities. Thus Moufang loops are algebras for a Lawvere theory, and thus can be defined internal to any category with finite products.

Moufang loops are power-associative, in that any bracketing of a string consisting of copies of the same element multiply to a unique element. In fact, more is true, in that any two elements generate a genuine group; that is, Moufang loops are alternative.

Examples

References

Related concepts in nnLab: quasigroup, Bol loop, composition algebra, identities of Bol-Moufang type