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—		moufang_loops [2010/07/29 15:46] (current)
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		+	=====Moufang loops=====
		+	Abbreviation: **MLoop**
		+	====Definition====
		+	A \emph{Moufang loop} is a [[loops]] $\mathbf{A}=\langle A,\cdot ,\backslash,/,e\rangle $ such that
		+
		+
		+	$((xy)z)x = x(y(zx))$, $y(x(yz)) = ((yx)y)z$, $(yx)(zy) = (y(xz))y$
		+
		+
		+	Remark:
		+
		+	==Morphisms==
		+	Let $\mathbf{A}$ and $\mathbf{B}$ be Moufang loops. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
		+
		+	$h(xy)=h(x)h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(e)=e$
		+
		+	====Examples====
		+	Example 1:
		+
		+	====Basic results====
		+
		+	====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 def. pr. cong.]]  | |
		+	^[[Amalgamation property]]  | |
		+	^[[Strong amalgamation property]]  | |
		+	^[[Epimorphisms are surjective]]  | |
		+	====Finite members====
		+
		+	$\begin{array}{lr}
		+	f(1)= &1\\
		+	f(2)= &\\
		+	f(3)= &\\
		+	f(4)= &\\
		+	f(5)= &\\
		+	f(6)= &\\
		+	f(7)= &\\
		+	\end{array}$
		+
		+	====Subclasses====
		+	[[Groups]]
		+
		+	====Superclasses====
		+	[[Loops]]
		+
		+
		+	====References====
		+
		+	[(Ln19xx>
		+	)]