Generalized MV-algebras

 
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 \begin{document}
 \textbf{\Large Generalized MV-algebras}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Generalized_MV-algebras}{edit}
 
 \abbreviation{GMV}
 
 \begin{definition}
 A \emph{generalized MV-algebra} is a \href{Residuated_lattices.pdf}{residuated lattices} 
 $\mathbf{L}=\langle L,\vee, \wedge, \cdot, e, \backslash, /\rangle$ such that
 
 $x\vee y=x/(y\backslash x\wedge e)$, $x\vee y=(x/y\wedge e)\backslash y$
 \end{definition}
 
 \begin{morphisms}
 Let $\mathbf{L}$ and $\mathbf{M}$ be generalized MV-algebras. A
 morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$
 that is a homomorphism: 
 
 $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)\ 
 \mbox{and}  h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash
 y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)\$, $
 h(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 [http://www.chapman.edu/~jipsen/lgroups/GMVDecisionProc.html implementation]\\\hline
 Quasiequational theory & \\\hline
 First-order theory & \\\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 & \\\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 $n$.
 
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 \end{finite_members}
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 \begin{subclasses}\ 
 
 
 \href{Commutative_generalized_MV-algebras.pdf}{Commutative generalized MV-algebras} 
 
 \href{Integral_generalized_MV-algebras.pdf}{Integral generalized MV-algebras} 
 
 \href{MV-algebras.pdf}{MV-algebras} 
 
 \end{subclasses}
 
 \begin{superclasses}\ 
 
 \href{Generalized_BL-algebras.pdf}{Generalized BL-algebras} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
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