Mathematical Structures: History of Lukasiewicz algebras of order n

# History of Lukasiewicz algebras of order n

 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \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 $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \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} %
 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \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 $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \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} %