Nonassociative relation algebras

http://mathcs.chapman.edu/structuresold/files/Nonassociative_relation_algebras.pdf

%%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 Nonassociative relation algebras}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Nonassociative_relation_algebras}{edit}

\abbreviation{NA}
\begin{definition}
A \emph{nonassociative relation algebra} is a structure $\mathbf{A}=\langle A,\vee,0,
\wedge,1,\neg,\circ,^{\smile},e\rangle$ such that


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


$e$ is an \emph{identity} for $\circ$:  $x\circ e=x$, $e\circ x=x$


$\circ$ is \emph{join-preserving}:  
$(x\vee y)\circ z=(x\circ z)\vee (y\circ z)$


$^{\smile}$ is an \emph{involution}:  
${x^\smile}^\smile=x$, $(x\circ y)^{\smile} z=y^{\smile}\circ x^{\smile}$


$^{\smile}$ is \emph{join-preserving}:  
$(x\vee y)^{\smile} z=x^{\smile}\vee y^{\smile}$


$\circ$ is residuated:  $x^{\smile}\circ(\neg (x\circ y))\le\neg y$


Remark: 

\end{definition}
\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be relation algebras. 
A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\circ$, $^{\smile}$, $e$:

$h(x\circ y)=h(x)\circ h(y)$, $h(x^{\smile})=h(x)^{\smile}$, $h(e)=e$
\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 & decidable\\\hline
Quasiequational theory & undecidable\\\hline
First-order theory & undecidable\\\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 & \\\hline
Equationally def. pr. cong. & \\\hline
Discriminator variety & 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 $n$.

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

\href{Weakly_associative_relation_algebras.pdf}{Weakly associative relation algebras} 

\end{subclasses}
\begin{superclasses}\ 

\href{Nonassociative_sequential_algebras.pdf}{Nonassociative sequential algebras} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
%