Allegories

http://math.chapman.edu/structuresold/files/Allegories.pdf

This is pdfTeX, Version 3.14159-1.10b (Web2C 7.4.5)
(./Allegories.tex{/usr/share/texmf/pdftex/config/pdftex.cfg}
LaTeX2e <2001/06/01>
Babel <v3.7h> and hyphenation patterns for american, french, german, ngerman, n
ohyphenation, loaded.
(/usr/share/texmf/tex/latex/amscls/amsart.cls
Document Class: amsart 2000/10/26 v2.08
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the '?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/hyperref/hyperref.sty
(/usr/share/texmf/tex/latex/graphics/keyval.sty)
(/usr/share/texmf/tex/latex/hyperref/pd1enc.def)
(/usr/share/texmf/tex/latex/config/hyperref.cfg)
Implicit mode ON; LaTeX internals redefined
(/usr/share/texmf/tex/latex/html/url.sty))
*hyperref using default driver hpdftex*
(/usr/share/texmf/tex/latex/hyperref/hpdftex.def
(/usr/share/texmf/tex/latex/psnfss/pifont.sty
(/usr/share/texmf/tex/latex/psnfss/upzd.fd)
(/usr/share/texmf/tex/latex/psnfss/upsy.fd))) (./Allegories.aux)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
(/usr/share/texmf/tex/latex/hyperref/nameref.sty) (./Allegories.out)
(./Allegories.out)

LaTeX Warning: `h' float specifier changed to `ht'.


LaTeX Warning: Empty `thebibliography' environment on input line 136.

[1{/usr/share/texmf/dvips/config/pdftex.map}] [2] (./Allegories.aux) ){/usr/sha
re/texmf/dvips/tetex/0ef0afca.enc}</usr/share/texmf/fonts/type1/bluesky/cm/cmcs
c10.pfb>{/usr/share/texmf/dvips/tetex/aae443f0.enc}</usr/share/texmf/fonts/type
1/bluesky/cm/cmmi8.pfb></usr/share/texmf/fonts/type1/bluesky/cm/cmmi12.pfb>{/us
r/share/texmf/dvips/tetex/bbad153f.enc}</usr/share/texmf/fonts/type1/bluesky/cm
/cmsy10.pfb>{/usr/share/texmf/dvips/tetex/74afc74c.enc}</usr/share/texmf/fonts/
type1/bluesky/cm/cmti12.pfb>{/usr/share/texmf/dvips/tetex/f7b6d320.enc}</usr/sh
are/texmf/fonts/type1/bluesky/cm/cmr12.pfb></usr/share/texmf/fonts/type1/bluesk
y/cm/cmbx12.pfb></usr/share/texmf/fonts/type1/bluesky/cm/cmr9.pfb>
Output written on Allegories.pdf (2 pages, 55423 bytes).
Transcript written on Allegories.log.

%%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}}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\pagestyle{myheadings}\thispagestyle{myheadings}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Allegories}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Allegories}{edit}

\abbreviation{All}

\begin{definition}
An \emph{allegory} is an expanded \href{Categories.pdf}{category} $\mathbf{M}=\langle M,\circ,\text{dom},\text{rng},\text{id},\vee,\wedge,^\smile\rangle$ such that

$...$ is ...:  $...$

$...$ is ...:  $...$

Remark: This is a template.
If you know something about allegories, click on the 'Edit text of this page' link at the bottom, then fill out some of the information.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be allegories. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a functor $F:A\rightarrow B$ that also preserves the
new operations: 
$h(x ... y)=h(x) ... h(y)$
\end{morphisms}

\begin{definition}
An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle
...\rangle$ such that

$...$ is ...:  $axiom$
  
$...$ is ...:  $axiom$
\end{definition}

\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})

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

\begin{tabular}{|ll|}\hline
  Classtype                       & partial algebras \\\hline
  Equational theory               & \\\hline
  Quasiequational theory          & \\\hline
  First-order theory              & \\\hline
  Locally finite                  & \\\hline
  Residual size                   & \\\hline
  Congruence distributive         & \\\hline
  Congruence modular              & \\\hline
  Congruence $n$-permutable       & \\\hline
  Congruence regular              & \\\hline
  Congruence uniform              & \\\hline
  Congruence extension property   & \\\hline
  Definable principal congruences & \\\hline
  Equationally def. pr. cong.     & \\\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 $n$.

$\begin{array}{lr}
  f(1)= &1\\
  f(2)= &\\
  f(3)= &\\
  f(4)= &\\
  f(5)= &\\
\end{array}$\qquad
$\begin{array}{lr}
  f(6)= &\\
  f(7)= &\\
  f(8)= &\\
  f(9)= &\\
  f(10)= &\\
\end{array}$

\end{finite_members}

\begin{subclasses}\ 

  \href{....pdf}{...} subvariety

  \href{....pdf}{...} expansion

\end{subclasses}

\begin{superclasses}\ 

  \href{....pdf}{...} supervariety

  \href{....pdf}{...} subreduct

\end{superclasses}

\begin{thebibliography}{10}

%\bibitem{Ln19xx}
%F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview} 

\end{thebibliography}

\end{document}
%