|
%%run pdflatex %
|
|
%%run pdflatex %
|
![[Home]](/structures.gif)
\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}
%