# Differences

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quasigroups [2010/07/29 15:46] (current)
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+=====Quasigroups=====
+Abbreviation: **Qgrp**
+====Definition====
+A \emph{quasigroup} is a structure $\mathbf{A}=\langle A,\cdot ,\backslash,/\rangle$ of type $\langle 2,2,2\rangle$ such that
+
+
+$(y/x)x = y$, $x(x\backslash y) = y$
+
+
+$(xy)/y = x$, $x\backslash(xy) = y$
+
+
+Remark:
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be quasigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
+
+$h(xy)=h(x)h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$.
+
+====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====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &1\\ +f(3)= &1\\ +f(4)= &5\\ +f(5)= &35\\ +f(6)= &1411\\ +f(7)= &1130531\\ +\end{array}$
+
+====Subclasses====
+[[Loops]]
+
+[[Medial quasigroups]]
+
+
+====Superclasses====
+
+====References====
+
+[(Ln19xx>
+)]

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