Boolean monoids

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

This is pdfTeX, Version 3.14159-1.10b (Web2C 7.4.5)
(./Boolean_monoids.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)))
No file Boolean_monoids.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) [1{/usr/share/texmf/dvips/con
fig/pdftex.map}] [2] (./Boolean_monoids.aux)

LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right.

 ){/usr/share/texmf/dvips/tetex/f7b6d320.enc}</usr/share/texmf/fonts/type1/blue
sky/cm/cmr10.pfb>{/usr/share/texmf/dvips/tetex/0ef0afca.enc}</usr/share/texmf/f
onts/type1/bluesky/cm/cmcsc10.pfb>{/usr/share/texmf/dvips/tetex/aae443f0.enc}</
usr/share/texmf/fonts/type1/bluesky/cm/cmmi12.pfb>{/usr/share/texmf/dvips/tetex
/bbad153f.enc}</usr/share/texmf/fonts/type1/bluesky/cm/cmsy10.pfb>{/usr/share/t
exmf/dvips/tetex/74afc74c.enc}</usr/share/texmf/fonts/type1/bluesky/cm/cmti12.p
fb></usr/share/texmf/fonts/type1/bluesky/cm/cmr12.pfb></usr/share/texmf/fonts/t
ype1/bluesky/cm/cmbx12.pfb></usr/share/texmf/fonts/type1/bluesky/cm/cmr9.pfb>
Output written on Boolean_monoids.pdf (2 pages, 54902 bytes).
Transcript written on Boolean_monoids.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}}
\pagestyle{myheadings}\thispagestyle{myheadings}
\markboth{\today}{math.chapman.edu/structures}

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

\abbreviation{BMon}
\begin{definition}
A \emph{Boolean monoid} is a structure $\mathbf{A}=\langle A,\vee,0,
\wedge,1,\neg,\cdot,e\rangle$ such that


$\langle A,\vee,0,
\wedge,1,\neg\rangle $ is a \href{Boolean_algebras.pdf}{Boolean algebra}


$\langle A,\cdot,e\rangle $ is a \href{Monoids.pdf}{monoids}


$\cdot$ is \emph{join-preserving} in each argument:  
$(x\vee y)\cdot z=(x\cdot z)\vee (y\cdot z) \mbox{ and } x\cdot (y\vee z)=(x\cdot y)\vee (x\cdot z)$


$\cdot$ is \emph{normal} in each argument:  $0\cdot x=0 \mbox{ and } x\cdot 0=0$


Remark: 

\end{definition}
\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean monoids. 
A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a Boolean homomorphism and preserves $\cdot$, $e$:

$h(x\cdot y)=h(x)\cdot h(y) \mbox{ and } h(e)=e$
\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 & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
Congruence distributive & yes\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, $n=2$\\\hline
Congruence regular & yes\\\hline
Congruence uniform & yes\\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & no\\\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 $n$.

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &0\\
f(4)= &9\\
f(5)= &0\\
f(6)= &0\\
f(7)= &0\\
f(8)= &258\\
\end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\ 

\href{Sequential_algebras.pdf}{Sequential algebras} 

\end{subclasses}
\begin{superclasses}\ 

\href{Boolean_algebras_with_operators.pdf}{Boolean algebras with operators} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
%