loops - MathStructures

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=====Loops=====

Abbreviation: **Loop**
====Definition====
A \emph{loop} is a structure $\mathbf{A}=\langle A,\cdot ,\backslash,/,e\rangle $ of type $\langle 2,2,2,0\rangle $ such that

$(y/x)x = y$, $x(x\backslash y) = y$

$(xy)/y = x$, $x\backslash(xy) = y$

$e$ is an identity for $\cdot$:  $xe = x$, $ex = x$

Remark: 

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be 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]]  |yes |
^[[Congruence n-permutable]]  |yes, $n=2$ |
^[[Congruence regular]]  | |
^[[Congruence uniform]]  | |
^[[Congruence extension property]]  | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &1\\
f(4)= &2\\
f(5)= &6\\
f(6)= &109\\
f(7)= &23746\\
f(8)= &106228849\\
f(9)= &9365022303540\\
f(10)= &20890436195945769617\\
f(11)= &1478157455158044452849321016\\
\end{array}$

====Subclasses====
[[Groups]] 

====Superclasses====
[[Quasigroups]] 


====References====

[(Ln19xx>
)]
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