Mathematical Structures: Groupoids

# Groupoids

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Changed: 1,113c1,113
 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Groupoids} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Groupoids}{edit} \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/} \parskip0pt \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} %
 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Groupoids} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Groupoids}{edit} \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/} \parskip0pt \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} %

http://mathcs.chapman.edu/structuresold/files/Groupoids.pdf
%%run pdflatex

%

\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\markboth{\today}{math.chapman.edu/structures}

\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/}
\parskip0pt
\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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