Mathematical Structures: Action algebras

# Action algebras

http://mathcs.chapman.edu/structuresold/files/Action_algebras.pdf
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\begin{document}
\textbf{\Large Action algebras}

\abbreviation{Act}
\begin{definition}
An \emph{action algebra} is a structure $\mathbf{A}=\langle A,\vee,\bot,\cdot,1,^*,\backslash,/\rangle$ of type
$\langle 2,0,2,0,1,2,2\rangle$ such that

$\langle A,\vee,\bot,\cdot,1,^*\rangle$ is a \href{Kleene_algebras.pdf}{Kleene algebras}

$\backslash$ is the left residual of $\cdot$:  $y\leq x\backslash z\Longleftrightarrow xy\leq z$

$/$ is the right residual of $\cdot$:  $x\leq z/y\Longleftrightarrow xy\leq z$

Remark:

\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be action algebras.
A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$,
$h(x/y)=h(x)/h(y)$, $h(x^*)=h(x)^*$, $h(\bot)=\bot$ and $h(1)=1$.
\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 \cite{Pratt1991}\\\hline
Equational theory & \\\hline
Quasiequational theory & undecidable\\\hline
First-order theory & undecidable\\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
Congruence distributive & yes \cite{AltRaf2004}\\\hline
Congruence modular & yes \\\hline
Congruence n-permutable & yes, $n=4$ \cite{AltRaf2004}\\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &3\\ f(4)= &20\\ f(5)= &149\\ f(6)= &1488\\ \end{array}$
\end{finite_members}

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\begin{subclasses}\

\href{Action_lattices.pdf}{Action lattices}

\end{subclasses}
\begin{superclasses}\

\href{Kleene_algebras.pdf}{Kleene algebras}

\href{Residuated_join-semilattices.pdf}{Residuated join-semilattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Pratt1991}
Vaughan Pratt, \emph{Action logic and pure induction},
Logics in AI (Amsterdam, 1990)'', Lecture Notes in Comput. Sci.,
478, 1991, 97--120, 92d:03016

\bibitem{AltRaf2004}
C.J. van Alten and J.G. Raftery, \emph{Embedding Theorems and Rule Separation in Logics without Weakening},
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