Mathematical Structures: Action lattices

# Action lattices

Showing revision 13
http://mathcs.chapman.edu/structuresold/files/Action_lattices.pdf
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\begin{document}

\textbf{\Large Action lattices}

\abbreviation{ActLat}

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

$\langle A,\vee,0,\cdot,1,^*\rangle$ is a \href{Kleene_algebras.pdf}{Kleene algebras}

$\langle A,\vee,\wedge\rangle$ is a \href{Lattices.pdf}{Lattices}

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

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

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be action lattices. A morphism from $\mathbf{A}$
to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y) \mbox{and} h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(x^*)=h(x)^*$, $h(0)=0$, $h(1)=1$
\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\\\hline
Equational theory & open\\\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 & no\\\hline
Congruence e-regular & yes\\\hline
Congruence uniform & no\\\hline
Congruence extension property & no\\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &3\\ f(4)= &16\\ f(5)= &149\\ f(6)= &1488\\ \end{array}$
\end{finite_members}

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

\href{Commutative_action_lattices.pdf}{Commutative action lattices}

\end{subclasses}

\begin{superclasses}\

\href{Action_algebras.pdf}{Action algebras}

\href{Residuated_lattices.pdf}{Residuated lattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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