Medial groupoids

 
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 \begin{document}
 \textbf{\Large Medial groupoids}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Medial_groupoids}{edit}
 
 \begin{definition}
 A \emph{medial groupoid} is a structure $\mathbf{G}=\langle G,\cdot\rangle$, where $\cdot $ is an infix binary operation such that
 
 
 $\cdot$ mediates:  $(x\cdot y)\cdot(z\cdot w)=(x\cdot z)\cdot (y\cdot w)$
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{G}$ and $\mathbf{H}$ be medial groupoids. A morphism from $\mathbf{G}$
 to $\mathbf{H}$ is a function $h:G\rightarrow H$ that is a homomorphism: 
 
 $h(xy)=h(x)h(y)$
 
 \end{morphisms}
 
 Jaroslav Jezek,Tomás Kepka,\emph{Equational theories of medial groupoids},
 Algebra Universalis,
 \textbf{17}1983,174--190\href{"http://www.ams.org/mathscinet-getitem?mr=85m:08011"}{MRreview}
 
 Jaroslav Jezek,Tomás Kepka,\emph{Medial groupoids},
 Rozpravy Ceskoslovenske Akad. Ved Rada Mat. Prirod. Ved,
 \textbf{93}1983,93\href{"http://www.ams.org/mathscinet-getitem?mr=85k:20165"}{MRreview}
 
 
 \begin{basic_results}
 \end{basic_results}
 \begin{examples}
 \begin{example}
 $\langle S,*\rangle$, where $\langle S,+,\cdot\rangle$ is any commutative semiring, $a,b\in S$, and $x*y=a\cdot x+b\cdot y$.
 \end{example}
 \end{examples}
 \begin{table}[h]
 \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})
 
 \begin{tabular}{|ll|}\hline
 Classtype & variety\\\hline
 Equational theory & \\\hline
 Quasiequational theory & \\\hline
 First-order theory & \\\hline
 Locally finite & no\\\hline
 Residual size & unbounded\\\hline
 Congruence distributive & no\\\hline
 Congruence modular & no\\\hline
 Congruence n-permutable & no\\\hline
 Congruence regular & no\\\hline
 Congruence uniform & no\\\hline
 Congruence extension property & \\\hline
 Definable principal congruences & \\\hline
 Equationally def. pr. cong. & no\\\hline
 Amalgamation property & \\\hline
 Strong amalgamation property & \\\hline
 Epimorphisms are surjective & \\\hline
 \end{tabular}
 \end{properties}
 \end{table}
 \begin{finite_members} $f(n)=$ number of members of size $n$.
 
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 f(1)= &1\\
 f(2)= &\\
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 \end{array}$
 \end{finite_members}
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 \begin{subclasses}\ 
 
 \href{Medial_semigroups.pdf}{Medial semigroups} 
 
 \href{Commutative_medial_groupoids.pdf}{Commutative medial groupoids} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Groupoids.pdf}{Groupoids} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
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 \end{document}
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