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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>
 +)]