# Moufang loops

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### Definition

A Moufang loop is a loop A = (A,·,\,/,e) such that
((xy)z)x  = x(y(zx))  and  y(x(yz))  = ((yx)y)z  and  (yx)(zy)  = (y(xz))y.

Remark:

### Morphisms

Let A and B be Moufang loops. A morphism from A to B is a function h : AB that is a homomorphism: h(xy) = h(x)h(y)  and  h(x\y) = h(x)\h(y)  and  h(x/y) = h(x)/h(y)  and  h(e) = e.

### Properties

 Classtype variety Equational theory decidable Quasiequational theory decidable First-order theory Locally finite no Residual size unbounded Congruence distributive no Congruence modular Congruence n-permutable Congruence regular Congruence uniform Congruence extension property Definable principal congruences Equationally definable principal congruences Amalgamation property Strong amalgamation property Epimorphisms are surjective

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Groups

### Superclasses

Loops

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