quasigroups - MathStructures

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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>
)]

Trace: • integral_ordered_monoids • commutative_lattice-ordered_rings • ordered_monoids • ordered_semilattices • commutative_residuated_partially_ordered_semigroups • hilbert_spaces • commutative_residuated_partially_ordered_monoids • relation_algebras • quandles • bilattices