Domain Theory Lecture Notes

] (b) edge[bend right] (tl) (tl) edge[loop above] (tl) (tr) edge[bend right=1.3em] (tl); \end {tikzpicture} } && && && {\begin {tikzpicture} \draw [rounded corners, thick, fill=white] (-0.8,-0.7) rectangle (0.8,1.39); \node (tr) at (0.5,0.5) {$\bullet $}; \node (tl) at (-0.5,0.5) {$\bullet $}; \node (b) at (0,-0.5) {$\bullet $}; \draw [thick] (b) -- (tr) (b) -- (tl); \draw [thick,draw=blue,->] (b) edge[bend right] (tr) (tl) edge[bend left=1.3em] (tr) (tr) edge[loop above] (tr); \end {tikzpicture} } \\ {\begin {tikzpicture} \draw [rounded corners, thick, fill=white] (-0.8,-1.2) rectangle (0.8,0.9); \node (tr) at (0.5,0.5) {$\bullet $}; \node (tl) at (-0.5,0.5) {$\bullet $}; \node (b) at (0,-0.5) {$\bullet $}; \draw [thick] (b) -- (tr) (b) -- (tl); \draw [thick,draw=blue,->] (b) edge[loop below] (b) (tl) edge[loop below] (tl) (tr) edge[bend right=1.3em] (tl); \end {tikzpicture} }\ar [u,ultra thick,-] && {\begin {tikzpicture} \draw [rounded corners, thick, fill=white] (-0.8,-1.2) rectangle (0.8,0.9); \node (tr) at (0.5,0.5) {$\bullet $}; \node (tl) at (-0.5,0.5) {$\bullet $}; \node (b) at (0,-0.5) {$\bullet $}; \draw [thick] (b) -- (tr) (b) -- (tl); \draw [thick,draw=blue] (b) edge[loop below,->] (b) (tl) edge[loop below,->] (tl) (tr) edge[loop below] (tr); \end {tikzpicture} } && {\begin {tikzpicture} \draw [rounded corners, thick, fill=white] (-0.8,-1.2) rectangle (0.8,0.9); \node (tr) at (0.5,0.5) {$\bullet $}; \node (tl) at (-0.5,0.5) {$\bullet $}; \node (b) at (0,-0.5) {$\bullet $}; \draw [thick] (b) -- (tr) (b) -- (tl); \draw [thick,draw=blue,->] (b) edge[loop below] (b) (tr) edge[bend left=1.3em] (tl) (tl) edge[bend left=1.3em] (tr); \end {tikzpicture} } && {\begin {tikzpicture} \draw [rounded corners, thick, fill=white] (-0.8,-1.2) rectangle (0.8,0.9); \node (tr) at (0.5,0.5) {$\bullet $}; \node (tl) at (-0.5,0.5) {$\bullet $}; \node (b) at (0,-0.5) {$\bullet $}; \draw [thick] (b) -- (tr) (b) -- (tl); \draw [thick,draw=blue,->] (b) edge[loop below] (b) (tl) edge[bend left=1.3em] (tr) (tr) edge[loop below] (tr); \end {tikzpicture} }\ar [u,ultra thick,-] \\ {\begin {tikzpicture} \draw [rounded corners, thick, fill=white] (-0.8,-1.2) rectangle (0.8,0.9); \node (tr) at (0.5,0.5) {$\bullet $}; \node (tl) at (-0.5,0.5) {$\bullet $}; \node (b) at (0,-0.5) {$\bullet $}; \draw [thick] (b) -- (tr) (b) -- (tl); \draw [thick,draw=blue,->] (b) edge[loop below] (b) (tl) edge[loop right] (tl) (tr) edge[bend left] (b); \end {tikzpicture} }\ar [u,ultra thick,-]\ar [urr,ultra thick,-] && {\begin {tikzpicture} \draw [rounded corners, thick, fill=white] (-0.8,-1.2) rectangle (0.8,0.9); \node (tr) at (0.5,0.5) {$\bullet $}; \node (tl) at (-0.5,0.5) {$\bullet $}; \node (b) at (0,-0.5) {$\bullet $}; \draw [thick] (b) -- (tr) (b) -- (tl); \draw [thick,draw=blue,->] (b) edge[loop below] (b) (tl) edge[bend right] (b) (tr) edge[bend right=1.3em] (tl); \end {tikzpicture} }\ar [ull,ultra thick,-]\ar [urr,ultra thick,-] && {\begin {tikzpicture} \draw [rounded corners, thick, fill=white] (-0.8,-1.2) rectangle (0.8,0.9); \node (tr) at (0.5,0.5) {$\bullet $}; \node (tl) at (-0.5,0.5) {$\bullet $}; \node (b) at (0,-0.5) {$\bullet $}; \draw [thick] (b) -- (tr) (b) -- (tl); \draw [thick,draw=blue,->] (b) edge[loop below] (b) (tr) edge[bend left] (b) (tl) edge[bend left=1.3em] (tr); \end {tikzpicture} }\ar [u,ultra thick,-]\ar [urr,ultra thick,-] && {\begin {tikzpicture} \draw [rounded corners, thick, fill=white] (-0.8,-1.2) rectangle (0.8,0.9); \node (tr) at (0.5,0.5) {$\bullet $}; \node (tl) at (-0.5,0.5) {$\bullet $}; \node (b) at (0,-0.5) {$\bullet $}; \draw [thick] (b) -- (tr) (b) -- (tl); \draw [thick,draw=blue,->] (b) edge[loop below] (b) (tl) edge[bend right] (b) (tr) edge[loop left] (tr); \end {tikzpicture} }\ar [u,ultra thick,-]\ar [ullll,ultra thick,-] \\ &\qquad &&{\begin {tikzpicture} \draw [rounded corners, thick, fill=white] (-0.8,-1.2) rectangle (0.8,0.9); \node (tr) at (0.5,0.5) {$\bullet $}; \node (tl) at (-0.5,0.5) {$\bullet $}; \node (b) at (0,-0.5) {$\bullet $}; \draw [thick] (b) -- (tr) (b) -- (tl); \draw [thick,draw=blue,->] (b) edge[loop below] (b) (tl) edge[bend right] (b) (tr) edge[bend left] (b); \end {tikzpicture} }\ar [ul, ultra thick,-] \ar [ur, ultra thick,-] \ar [ulll, ultra thick,-] \ar [urrr, ultra thick,-] &&\qquad \end {tikzcd} ]]> At each step "upwards" on this , one arrow that ends in now points to a non- value (moving up the ).