Complete semilattices

 
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 \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 Complete semilattices}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Complete_semilattices}{edit}
 
 \abbreviation{CSlat}
 \begin{definition}
 A \emph{complete semilattice} is a \href{Directed_complete_partial_orders.pdf}{directed complete partial orders} $\mathbf{P}=\langle P,\leq \rangle $
 such that every nonempty subset of $P$ has a greatest lower bound: 
 $\forall S\subseteq P\ (S\ne\emptyset\implies \exists z\in P(z=\bigwedge S))$.
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{P}$ and $\mathbf{Q}$ be complete semilattices. A morphism from $\mathbf{P}$ to 
 $\mathbf{Q}$ is a function $f:P\rightarrow Q$ that preserves all nonempty meets and all directed joins: 
 
 $z=\bigwedge S\implies f(z)=\bigwedge f[S]$ for all nonempty $S\subseteq P$ and 
 $z=\bigvee D\implies f(z)= \bigvee f[D]$
 
 \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 & second-order\\\hline
 Amalgamation property & \\\hline
 Strong amalgamation property & \\\hline
 Epimorphisms are surjective & \\\hline
 \end{tabular}
 \end{properties}
 \end{table}
 \begin{finite_members} $f(n)=$ number of members of size $n$.
 
 $\begin{array}{lr}
 f(1)= &1\\
 f(2)= &\\
 f(3)= &\\
 f(4)= &\\
 f(5)= &\\
 f(6)= &\\
 \end{array}$
 \end{finite_members}
 \hyperbaseurl{http://math.chapman.edu/structures/files/}
 \parskip0pt
 \begin{subclasses}\ 
 
 \href{Complete_lattices.pdf}{Complete lattices} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Directed_complete_partial_orders.pdf}{Directed complete partial orders} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
 %

 
 \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 Complete semilattices}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Complete_semilattices}{edit}
 
 \abbreviation{CSlat}
 \begin{definition}
 A \emph{complete semilattice} is a \href{Directed_complete_partial_orders.pdf}{directed complete partial orders} $\mathbf{P}=\langle P,\leq \rangle $
 such that every nonempty subset of $P$ has a greatest lower bound: 
 $\forall S\subseteq P\ (S\ne\emptyset\implies \exists z\in P(z=\bigwedge S))$.
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{P}$ and $\mathbf{Q}$ be complete semilattices. A morphism from $\mathbf{P}$ to 
 $\mathbf{Q}$ is a function $f:P\rightarrow Q$ that preserves all nonempty meets and all directed joins: 
 
 $z=\bigwedge S\implies f(z)=\bigwedge f[S]$ for all nonempty $S\subseteq P$ and 
 $z=\bigvee D\implies f(z)= \bigvee f[D]$
 
 \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 & second-order\\\hline
 Amalgamation property & \\\hline
 Strong amalgamation property & \\\hline
 Epimorphisms are surjective & \\\hline
 \end{tabular}
 \end{properties}
 \end{table}
 \begin{finite_members} $f(n)=$ number of members of size $n$.
 
 $\begin{array}{lr}
 f(1)= &1\\
 f(2)= &\\
 f(3)= &\\
 f(4)= &\\
 f(5)= &\\
 f(6)= &\\
 \end{array}$
 \end{finite_members}
 \hyperbaseurl{http://math.chapman.edu/structures/files/}
 \parskip0pt
 \begin{subclasses}\ 
 
 \href{Complete_lattices.pdf}{Complete lattices} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Directed_complete_partial_orders.pdf}{Directed complete partial orders} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
 %

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

%%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 Complete semilattices}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Complete_semilattices}{edit}

\abbreviation{CSlat}
\begin{definition}
A \emph{complete semilattice} is a \href{Directed_complete_partial_orders.pdf}{directed complete partial orders} $\mathbf{P}=\langle P,\leq \rangle $
such that every nonempty subset of $P$ has a greatest lower bound: 
$\forall S\subseteq P\ (S\ne\emptyset\implies \exists z\in P(z=\bigwedge S))$.
\end{definition}
\begin{morphisms}
Let $\mathbf{P}$ and $\mathbf{Q}$ be complete semilattices. A morphism from $\mathbf{P}$ to 
$\mathbf{Q}$ is a function $f:P\rightarrow Q$ that preserves all nonempty meets and all directed joins: 

$z=\bigwedge S\implies f(z)=\bigwedge f[S]$ for all nonempty $S\subseteq P$ and 
$z=\bigvee D\implies f(z)= \bigvee f[D]$

\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 & second-order\\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr}
f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
\end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\ 

\href{Complete_lattices.pdf}{Complete lattices} 

\end{subclasses}
\begin{superclasses}\ 

\href{Directed_complete_partial_orders.pdf}{Directed complete partial orders} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
%