\documentclass[12,landscape]{amsart}
\usepackage{tikz}
\topmargin-.5in
\oddsidemargin-0.5in
\textheight8in
\textwidth10in
\thispagestyle{empty}
\begin{document}
\begin{center}\Huge\bf
A distributive lattice with non-modular MacNeille completion
\end{center}

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\begin{center}
\qquad
\begin{tikzpicture}[scale=.8, every node/.style={draw,fill=white,circle,inner sep=2}]
\draw[dotted,thick](1.5,1.5)--(2,2)(0.5,2.5)--(1,3)(-0.5,2.5)--(-1,3)(-1.5,1.5)--(-2,2);
\draw(0,0)node{}--(1,1)node{}--(0,2)node{}--(-1,1)node{}--(0,0)
(1,1)--(1.5,1.5)(0,2)--(0.5,2.5)(0,2)--(-0.5,2.5)(-1,1)--(-1.5,1.5);
\draw[dotted,thick](4.5,4.5)--(4,4)(3.5,5.5)--(3,5)(3.5,6.5)--(3,7)(4.5,7.5)--(4,8);
\draw(5,5)--(4.5,4.5)(4,6)--(3.5,6.5)(4,6)--(3.5,5.5)(5,7)--(4.5,7.5)
(5,5)node{}--(6,6)node{}--(5,7)node{}--(4,6)node{}--(5,5)
(5,5)--(5,6)(6,6)--(6,7)(5,7)--(5,8)(4,6)--(4,7);
\draw[dotted,thick](4.5,5.5)--(4,5)(3.5,6.5)--(3,6)(3.5,7.5)--(3,8)(4.5,8.5)--(4,9);
\draw(5,6)--(4.5,5.5)(4,7)--(3.5,7.5)(4,7)--(3.5,6.5)(5,8)--(4.5,8.5);
\draw(5,6)node{}--(6,7)node{}--(5,8)node{}--(4,7)node{}--(5,6);
\draw[dotted,thick](2,11)--(1.5,11.5)(1,10)--(0.5,10.5)(-1,10)--(-0.5,10.5)
(-2,11)--(-1.5,11.5);
\draw(1,12)--(1.5,11.5)(0,11)--(0.5,10.5)(0,11)--(-0.5,10.5)(-1,12)--(-1.5,11.5);
\draw(0,11)node{}--(1,12)node{}--(0,13)node{}--(-1,12)node{}--(0,11);
\node at(0,-2)[draw=white]{\huge$L$};
\end{tikzpicture}
\qquad\qquad
%
\begin{tikzpicture}[scale=.8, every node/.style={draw,fill=white,circle,inner sep=2}]
\draw[dotted,thick](1.5,1.5)--(2,2)(0.5,2.5)--(1,3)(-0.5,2.5)--(-1,3)(-1.5,1.5)--(-2,2);
\draw(0,0)node{}--(1,1)node{}--(0,2)node{}--(-1,1)node{}--(0,0)
(1,1)--(1.5,1.5)(0,2)--(0.5,2.5)(0,2)--(-0.5,2.5)(-1,1)--(-1.5,1.5);
\draw[dotted,thick](4.5,4.5)--(4,4)(3.5,5.5)--(3,5)(3.5,6.5)--(3,7)(4.5,7.5)--(4,8);
\draw(5,5)--(4.5,4.5)(4,6)--(3.5,6.5)(4,6)--(3.5,5.5)(5,7)--(4.5,7.5)
(5,5)node{}--(6,6)node[black]{}--(5,7)node{}--(4,6)node{}--(5,5)
(5,5)--(5,6)(6,6)--(6,7)(5,7)--(5,8)(4,6)--(4,7);
\draw[dotted,thick](4.5,5.5)--(4,5)(3.5,6.5)--(3,6)(3.5,7.5)--(3,8)(4.5,8.5)--(4,9);
\draw(5,6)--(4.5,5.5)(4,7)--(3.5,7.5)(4,7)--(3.5,6.5)(5,8)--(4.5,8.5);
\draw(5,6)node{}--(6,7)node[black]{}--(5,8)node{}--(4,7)node{}--(5,6);
\draw[dotted,thick](2,11)--(1.5,11.5)(1,10)--(0.5,10.5)(-1,10)--(-0.5,10.5)
(-2,11)--(-1.5,11.5)(1.5,4.5)--(1,5)(1.5,8.5)--(1,8);
\draw(1,12)--(1.5,11.5)(0,11)--(0.5,10.5)(0,11)--(-0.5,10.5)(-1,12)--(-1.5,11.5);
\draw(0,11)node{}--(1,12)node{}--(0,13)node{}--(-1,12)node{}--(0,11);
\draw(3,3)node[black]{}--(2,4)node{}--(1.5,4.5)(3,10)node[black]{}--(2,9)node{}--(1.5,8.5)
(0,6.5)node[black]{};
\node at(0,-2)[draw=white]{\huge$\overline L$};
\end{tikzpicture}
\qquad\qquad
%
\begin{tikzpicture}[scale=.8, every node/.style={draw,fill=white,circle,inner sep=2}]
\draw[dotted,thick](1.5,1.5)--(2,2)(0.5,2.5)--(1,3)(-0.5,2.5)--(-1,3)(-1.5,1.5)--(-2,2);
\draw(0,0)node{}--(1,1)node{}--(0,2)node{}--(-1,1)node{}--(0,0)
(1,1)--(1.5,1.5)(0,2)--(0.5,2.5)(0,2)--(-0.5,2.5)(-1,1)--(-1.5,1.5);
\draw[dotted,thick](4.5,4.5)--(4,4)(3.5,5.5)--(3,5)(3.5,6.5)--(3,7)(4.5,7.5)--(4,8);
\draw(5,5)--(4.5,4.5)(4,6)--(3.5,6.5)(4,6)--(3.5,5.5)(5,7)--(4.5,7.5)
(5,5)node{}--(6,6)node{}--(5,7)node{}--(4,6)node{}--(5,5)
(5,5)--(5,6)(6,6)--(6,7)(5,7)--(5,8)(4,6)--(4,7);
\draw[dotted,thick](4.5,5.5)--(4,5)(3.5,6.5)--(3,6)(3.5,7.5)--(3,8)(4.5,8.5)--(4,9);
\draw(5,6)--(4.5,5.5)(4,7)--(3.5,7.5)(4,7)--(3.5,6.5)(5,8)--(4.5,8.5);
\draw(5,6)node{}--(6,7)node{}--(5,8)node{}--(4,7)node{}--(5,6);
\draw[dotted,thick](2,11)--(1.5,11.5)(1,10)--(0.5,10.5)(-1,10)--(-0.5,10.5)
(-2,11)--(-1.5,11.5)(1.25,4.25)--(0.75,4.75)(1.25,8.75)--(0.75,8.25)
(1.75,4.75)--(1.25,5.25)(1.75,8.25)--(1.25,7.75)
(-1.25,4.25)--(-0.75,4.75)
(-1.25,8.75)--(-0.75,8.25)
(1.75,5.75)--(1.25,6.25)
(1.75,7.25)--(1.25,6.75);
\draw(1,12)--(1.5,11.5)(0,11)--(0.5,10.5)(0,11)--(-0.5,10.5)(-1,12)--(-1.5,11.5);
\draw(0,11)node{}--(1,12)node{}--(0,13)node{}--(-1,12)node{}--(0,11);
\draw(3.25,3.25)--(2.75,2.75)node[blue]{}--(1.75,3.75)node[blue]{}--(1.25,4.25)
(3.25,9.75)--(2.75,10.25)node[red]{}--(1.75,9.25)node[red]{}--(1.25,8.75)
(3.25,3.25)node[red]{}--(2.25,4.25)node[red]{}--(1.75,4.75)(2.25,4.25)--(1.75,3.75)
(3.25,9.75)node[blue]{}--(2.25,8.75)node[blue]{}--(1.75,8.25)(2.25,8.75)--(1.75,9.25)
(-2.75,2.75)node[blue]{}--(-1.75,3.75)node[blue]{}--(-1.25,4.25)
(-2.75,10.25)node[red]{}--(-1.75,9.25)node[red]{}--(-1.25,8.75)
(3.25,3.25)--(3.25,4.25)node[red]{}--(2.25,5.25)node[red]{}--(1.75,5.75)(2.25,5.25)--(2.25,4.25)
(3.25,9.75)--(3.25,8.75)node[blue]{}--(2.25,7.75)node[blue]{}--(1.75,7.25)(2.25,7.75)--(2.25,8.75)
(0,5.5)node[blue]{}--(.5,6)node[green]{}--(.5,7)node[green]{}--(0,7.5)node[red]{};
\node at(0,-2)[draw=white]{\huge$L^\delta$};
\end{tikzpicture}
\end{center}

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\begin{center}\Large
The middle example is due to N. Funayama [1944].
Note that the canonical extension $L^\delta$ of $L$ is still 
distributive. \\
The blue elements are in $O(L^\delta)$, the red elements are in
$K(L^\delta)$ and the green elements are neither open nor closed.
\\[10pt]
Peter Jipsen --- February 2016 --- Chapman University
\end{center}
\end{document}
