Table of Contents
Quasigroups
Abbreviation: Qgrp
Definition
A 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
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
Superclasses
References
Trace: » quasigroups