Moufang loops

Abbreviation: MLoop

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 : A→B 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.

Some results

Examples

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

Finite members

[Size 1]?: 1
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Subclasses

Groups

Superclasses

Loops