Mathematical Structures: Medial groupoids

# Medial groupoids

http://mathcs.chapman.edu/structuresold/files/Medial_groupoids.pdf
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\begin{document}
\textbf{\Large Medial groupoids}

\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, Tomas 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, Tomas Kepka,\emph{Medial groupoids},
\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 $n$.

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

\end{thebibliography}

\end{document}
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