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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> | ||
+ | )] |
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