Mathematical Structures: History of Distributive p-algebras

# History of Distributive p-algebras

 %%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 Distributive p-algebras} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Distributive_p-algebras}{edit} \abbreviation{DpAlg} \begin{definition} A \emph{distributive p-algebra} is a structure $\mathbf{L}=\langle L,\vee ,0,\wedge ,1,^*\rangle$ such that $\langle L,\vee,0,\wedge,1\rangle$ is a \href{Bounded_distributive_lattices.pdf}{bounded distributive lattices} $x^*$ is the \emph{pseudo complement} of $x$: $y\leq x^* \iff x\wedge y=0$ \end{definition} \begin{morphisms} Let $\mathbf{L}$ and $\mathbf{M}$ be distributive p-algebras. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism: $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(1)=1$, $h(x^*)=h(x)^*$ \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 & yes\\\hline Strong amalgamation property & \\\hline Epimorphisms are surjective & \\\hline Locally finite & \\\hline Residual size & \\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Distributive_double_p-algebras.pdf}{Distributive double p-algebras} \end{subclasses} \begin{superclasses}\ \href{Distributive_lattices.pdf}{Distributive lattices} \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 Distributive p-algebras} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Distributive_p-algebras}{edit} \abbreviation{DpAlg} \begin{definition} A \emph{distributive p-algebra} is a structure $\mathbf{L}=\langle L,\vee ,0,\wedge ,1,^*\rangle$ such that $\langle L,\vee,0,\wedge,1\rangle$ is a \href{Bounded_distributive_lattices.pdf}{bounded distributive lattices} $x^*$ is the \emph{pseudo complement} of $x$: $y\leq x^* \iff x\wedge y=0$ \end{definition} \begin{morphisms} Let $\mathbf{L}$ and $\mathbf{M}$ be distributive p-algebras. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism: $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(1)=1$, $h(x^*)=h(x)^*$ \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 & yes\\\hline Strong amalgamation property & \\\hline Epimorphisms are surjective & \\\hline Locally finite & \\\hline Residual size & \\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Distributive_double_p-algebras.pdf}{Distributive double p-algebras} \end{subclasses} \begin{superclasses}\ \href{Distributive_lattices.pdf}{Distributive lattices} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %