right_quasigroups - MathStructures

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=====Right quasigroups=====

Abbreviation: **RQgrp**
====Definition====
A \emph{right quasigroup} is a structure $\mathbf{A}=\langle A,\cdot,/\rangle$ of type $\langle 2,2\rangle $ such that

$(y/x)x = y$

$(xy)/y = x$

Remark: 

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be right quasigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: 

$h(xy)=h(x)h(y)$, $h(x/y)=h(x)/h(y)$.

====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====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &3\\
f(3)= &44\\
f(4)= &14022\\
f(5)= &\\
f(6)= &\\
f(7)= &\\
\end{array}$

See http://oeis.org/A193623

====Subclasses====
[[Right loops]] 

[[Quasigroups]]


====Superclasses====

====References====

[(Ln19xx>
)]

Trace: • equivalence_algebras • dokuwiki • multiplicative_additive_linear_logic_algebras • intuitionistic_linear_logic_algebras

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