Relation algebras

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

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

\abbreviation{RA}
\begin{definition}
A \emph{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}

$\langle A,\circ,e\rangle $ is a \href{Monoids.pdf}{monoid}

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

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

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

$\circ$ is residuated: $x^{\smile}\circ(\neg (x\circ y))\le\neg y$
\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 & undecidable\\\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 & yes\\\hline
Equationally def. pr. cong. & yes\\\hline
Discriminator variety & yes\\\hline
Amalgamation property & no\\\hline
Strong amalgamation property & no\\\hline
Epimorphisms are surjective & no\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

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[http://localhost/gap/ramaddux.html Small relation algebras]

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\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{subclasses}\ 

\href{n-dimensional_relation_algebras.pdf}{n-dimensional relation algebras} 

\href{Representable_relation_algebras.pdf}{Representable relation algebras} 

\href{Commutative_relation_algebras.pdf}{Commutative relation algebras} 

\href{Square-increasing_relation_algebras.pdf}{Square-increasing relation algebras} 

\end{subclasses}
\begin{superclasses}\ 

\href{Sequential_algebras.pdf}{Sequential algebras} 

\href{Semiassociative_relation_algebras.pdf}{Semiassociative relation algebras} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

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\end{document}
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