Mathematical Structures: Groupoids

# Groupoids

http://mathcs.chapman.edu/structuresold/files/Groupoids.pdf
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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Groupoids}

\abbreviation{BinOp}
\begin{definition}
A \emph{groupoid} is a structure $\mathbf{A}=\langle A,\cdot\rangle$ where
$\cdot$ is any binary operation on $A$.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be groupoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(x\cdot y)=h(x)\cdot h(y)$
\end{morphisms}

\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
\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 &   decidable\\\hline
Quasiequational theory &   \\\hline
First-order theory &   undecidable\\\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 &   no\\\hline
Definable principal congruences &   no\\\hline
Equationally def. pr. cong. &   no\\\hline
Amalgamation property &   yes\\\hline
Strong amalgamation property &   yes\\\hline
Epimorphisms are surjective &   yes\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &10\\ f(3)= &3330\\ f(4)= &178981952\\ f(5)= &2483527537094825\\ f(6)= &14325590003318891522275680\\ f(7)= &50976900301814584087291487087214170039\\ f(8)= &155682086691137947272042502251643461917498835481022016\\ Michael A. Harrison,\emph{The number of isomorphism types of finite algebras}, Proc. Amer. Math. Soc., \textbf{17}1966,731--737\href{"http://www.ams.org/mathscinet-getitem?mr=34 :118"}{MRreview} \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{subclasses}\

\href{Commutative_groupoids.pdf}{Commutative groupoids}

\href{Idempotent_groupoids.pdf}{Idempotent groupoids}

\href{Semigroups.pdf}{Semigroups}

\href{Left-distributive_groupoids.pdf}{Left-distributive groupoids}

\end{subclasses}
\begin{superclasses}\

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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