Mathematical Structures: Cancellative residuated lattices

Cancellative residuated lattices

http://mathcs.chapman.edu/structuresold/files/Cancellative_residuated_lattices.pdf
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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Cancellative residuated lattices}

\abbreviation{CanRL}
\begin{definition}
A \emph{cancellative residuated lattice} is a
\href{Residuated_lattices.pdf}{residuated lattice}
$\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ such that

$\cdot$ is right-cancellative:  $xz=yz\implies x=y$

$\cdot$ is left-cancellative:  $zx=zy\implies x=y$
\end{definition}

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

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$
$h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$ and $h(e)=e$
\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 & \\\hline
Quasiequational theory & \\\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 $n$.

$\begin{array}{lr} None \end{array}$
\end{finite_members}

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

\href{Cancellative_commutative_residuated_lattices.pdf}{Cancellative commutative residuated lattices}

\href{Cancellative_distributive_residuated_lattices.pdf}{Cancellative distributive residuated lattices}

\end{subclasses}
\begin{superclasses}\

\href{Residuated_lattices.pdf}{Residuated lattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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