Mathematical Structures: Nonassociative relation algebras

# Nonassociative relation algebras

http://mathcs.chapman.edu/structuresold/files/Nonassociative_relation_algebras.pdf
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\begin{document}
\textbf{\Large Nonassociative relation algebras}

\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}
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\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}

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\end{document}
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