History of Lukasiewicz algebras of order n

 
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 \begin{document}
 \textbf{\Large Lukasiewicz algebras of order n}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Lukasiewicz_algebras_of_order_n}{edit}
 
 \abbreviation{LA$_n$}
 \begin{definition}
 A \emph{Lukasiewicz algebra of order $n$} is a structure $\mathbf{A}=\langle A,\vee
 ,0,\wedge ,1,\neg,\sigma_0,\ldots,\sigma_{n-1}\rangle $ such that
 
 
 $\langle A,\vee ,0,\wedge ,1, \neg\rangle $ is a \href{De_Morgan_algebras.pdf}{De Morgan algebras}
 
 
 1. 
 $\sigma_i$ is a lattice homomorphism:  $\sigma_i(x\vee y)=\sigma_i(x)\vee\sigma_i(y)
 \mbox{and} \sigma_i(x\wedge y)=\sigma_i(x)\wedge\sigma_i(y)$
 
 2. 
 $\sigma_i(x) \vee \neg(\sigma_i(x)) = 1$, $\sigma_i(x) \wedge \neg(\sigma_i(x)) = 0$
 
 3. 
 $\sigma_i(\sigma_j(x)) = \sigma_j(x)$ for $1 \le j \le n-1$
 
 4. 
 $\sigma_i(\neg x) = \neg(\sigma_{n-i}(x))$
 
 5. 
 $\sigma_i(x) \wedge \sigma_j(x) = \sigma_i(x)$ for $i \le j \le n - 1$
 
 6. 
 $x \vee \sigma_{n-1}(x) = \sigma_{n-1}(x)$, $x \wedge \sigma_1(x) = \sigma_1(x)$
 
 7. 
 $y \wedge (x \vee \neg(\sigma_i(x)) \vee \sigma_{i+1}(y)) = y$ for $i \ne n - 1$
 \end{definition}
 
 \begin{morphisms}
 Let $\mathbf{A}$ and $\mathbf{B}$ be Lukasiewicz algebras of order $n$. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
 homomorphism: 
 
 $h(x\vee y)=h(x)\vee h(y)$, $h(\neg x)=\neg h(x)$, $h(\sigma_i(x))=\sigma_i(h(x))$ for $i=0,\ldots,n-1$
 
 \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 & decidable\\\hline
 Quasiequational theory & \\\hline
 First-order theory & \\\hline
 Congruence distributive & Yes\\\hline
 Congruence modular & Yes\\\hline
 Congruence n-permutable & \\\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
 Locally finite & yes\\\hline
 Residual size & $n$\\\hline
 \end{tabular}
 \end{properties}
 \end{table}
 \begin{finite_members} $f(n)=$ number of members of size $n$.
 
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 \begin{subclasses}\ 
 
 \href{Boolean_algebras.pdf}{Boolean algebras} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{De_Morgan_algebras.pdf}{De Morgan algebras} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
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