diff --git a/frap_book.tex b/frap_book.tex index 8360ecd..24f7249 100644 --- a/frap_book.tex +++ b/frap_book.tex @@ -1793,27 +1793,27 @@ As another example, consider a lattice tracking prime factors of numbers, up to Then the picture version might go like so: \begin{center}\begin{tikzpicture}[node distance=1.5cm] -\node(top) {$\{2, 3, 5\}$}; -\node(twothree) [below left of=top] {$\{2, 3\}$}; -\node(twofive) [below of=top] {$\{2, 5\}$}; -\node(threefive) [below right of=top] {$\{3, 5\}$}; -\node(two) [below of=twothree] {$\{2\}$}; -\node(three) [below of=twofive] {$\{3\}$}; -\node(five) [below of=threefive] {$\{5\}$}; -\node(emp) [below of=three] {$\{\}$}; +\node(top) {$\{\}$}; +\node(two) [below left of=top] {$\{2\}$}; +\node(three) [below of=top] {$\{3\}$}; +\node(five) [below right of=top] {$\{5\}$}; +\node(twothree) [below left of=two] {$\{2, 3\}$}; +\node(twofive) [below of=three] {$\{2, 5\}$}; +\node(threefive) [below right of=five] {$\{3, 5\}$}; +\node(bot) [below of=twofive] {$\{2, 3, 5\}$}; -\draw(top) -- (twothree); -\draw(top) -- (twofive); -\draw(top) -- (threefive); -\draw(twothree) -- (two); -\draw(twothree) -- (three); -\draw(twofive) -- (two); -\draw(twofive) -- (five); -\draw(threefive) -- (three); -\draw(threefive) -- (five); -\draw(two) -- (emp); -\draw(three) -- (emp); -\draw(five) -- (emp); +\draw(top) -- (two); +\draw(top) -- (three); +\draw(top) -- (five); +\draw(two) -- (twothree); +\draw(two) -- (twofive); +\draw(three) -- (twothree); +\draw(three) -- (threefive); +\draw(five) -- (twofive); +\draw(five) -- (threefive); +\draw(twothree) -- (bot); +\draw(twofive) -- (bot); +\draw(threefive) -- (bot); \end{tikzpicture}\end{center} Since $\sqsubseteq$ is clearly transitive, upward-moving paths across multiple nodes also imply $\sqsubseteq$ relationships between their endpoints.