Mathematical Structures: Lukasiewicz algebras of order n

# Lukasiewicz algebras of order n

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http://mathcs.chapman.edu/structuresold/files/Lukasiewicz_algebras_of_order_n.pdf
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\begin{document}
\textbf{\Large Lukasiewicz algebras of order n}

\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$
\end{finite_members}
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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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