\begin{frame} \small \vspace{-1ex} \begin{definition} Let $\AA = \langle A, \to_1,\to_2 \rangle$ be an ARS. \begin{itemize} \item \alert<1-2>{$\to_1$ commutes with $\to_2$} \begin{itemize} \item ${\FromBP{2}{*} \cdot \to_1^*} \:\subseteq\: {\to_1^* \cdot \FromBP{2}{*}}$ \smallskip \item<2-> \hspace{-1mm}\begin{tikzpicture}[on grid,baseline=(1).baseline] \node (1) {$\strut a$}; \node (2) [below left=of 1] {$b$}; \node (3) [below right=of 1] {$c$}; \node (4) [below right=of 2] {$\strut d$}; \node (5) [left=of 1] {\makebox[3cm][l]{$\strut\forall a,b,c$}}; \node (6) [left=of 4] {\makebox[3cm][l]{$\strut\exists d$}}; \draw[->>,blue!75!black] (1) -- (2) node [midway,above left] {$2$}; \draw[->>,green!50!black] (1) -- (3) node [midway,above right] {$1$}; \draw[dotted,->>,green!50!black] (2) -- (4) node [midway,below left] {$1$}; \draw[dotted,->>,blue!75!black] (3) -- (4) node [midway,below right] {$2$}; \end{tikzpicture} \end{itemize} \item<3-> \alert<3-4>{$\to_1$ commutes weakly with $\to_2$} \begin{itemize} \item ${\FromBP{2}{} \cdot \to_1} \:\subseteq\: {\to_1^* \cdot \FromBP{2}{*}}$ \smallskip \item<4-> \hspace{-1mm}\begin{tikzpicture}[on grid,baseline=(1).baseline] \node (1) {$\strut a$}; \node (2) [below left=of 1] {$b$}; \node (3) [below right=of 1] {$c$}; \node (4) [below right=of 2] {$\strut d$}; \node (5) [left=of 1] {\makebox[3cm][l]{$\strut\forall a,b,c$}}; \node (6) [left=of 4] {\makebox[3cm][l]{$\strut\exists d$}}; \draw[->,blue!75!black] (1) -- (2) node [midway,above left] {$2$}; \draw[->,green!50!black] (1) -- (3) node [midway,above right] {$1$}; \draw[dotted,->>,green!50!black] (2) -- (4) node [midway,below left] {$1$}; \draw[dotted,->>,blue!75!black] (3) -- (4) node [midway,below right] {$2$}; \end{tikzpicture} \end{itemize} \end{itemize} \end{definition} \end{frame}