Mathematical Structures: Pseudocomplemented distributive lattices

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\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
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\begin{document}
\textbf{\Large Pseudocomplemented distributive lattices}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Pseudocomplemented_distributive_lattices}{edit}

\abbreviation{pcDLat}

\begin{definition}
A \emph{pseudocomplemented distributive lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,^*\rangle$ such that

$\langle L,\vee,0,\wedge\rangle $ is a \href{Distributive_lattices.pdf}{distributive lattices} with bottom element $0$

$x^*$ 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 pseudocomplemented distributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(x^*)=h(x)^*$
\end{morphisms}

\begin{definition}
A \emph{pseudocomplemented distributive lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,^*\rangle$ such that

$\langle L,\vee,0,\wedge\rangle $ is a \href{Distributive_lattices.pdf}{distributive lattices}

$0$ is the bottom element: $0\leq x$

$x\wedge(x\wedge y)^*=x\wedge y^*$

$x\wedge 0^*=x$

$0^{**}=0$
\end{definition}

\begin{basic_results}
Pseudocomplemented distributive lattices are term equivalent to \href{Distributive_p-algebras.pdf}{distributive p-algebras}.
\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}
%

http://mathcs.chapman.edu/structuresold/files/Pseudocomplemented_distributive_lattices.pdf

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\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 Pseudocomplemented distributive lattices}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Pseudocomplemented_distributive_lattices}{edit}

\abbreviation{pcDLat}

\begin{definition}
A \emph{pseudocomplemented distributive lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,^*\rangle$ such that


$\langle L,\vee,0,\wedge\rangle $ is a \href{Distributive_lattices.pdf}{distributive lattices} with bottom element $0$


$x^*$ 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 pseudocomplemented distributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism: 

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(x^*)=h(x)^*$
\end{morphisms}

\begin{definition}
A \emph{pseudocomplemented distributive lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,^*\rangle$ such that


$\langle L,\vee,0,\wedge\rangle $ is a \href{Distributive_lattices.pdf}{distributive lattices}


$0$ is the bottom element:  $0\leq x$


$x\wedge(x\wedge y)^*=x\wedge y^*$


$x\wedge 0^*=x$


$0^{**}=0$
\end{definition}


\begin{basic_results}
Pseudocomplemented distributive lattices are term equivalent to \href{Distributive_p-algebras.pdf}{distributive p-algebras}.
\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}
%