\begin{frame} \small \begin{definition} \smallskip A property of TRSs is \alert{modular} if it is preserved under union. \smallskip \end{definition} \medskip \begin{remark}<2-> \smallskip Without further restrictions `no' property of TRSs is modular: \onslide<3-> \[ \begin{tabular}{l@{\qquad\qquad}ll} termination & \GREEN{$\begin{array}{r@{~}c@{~}l} \m{a} & \to & \m{b} \end{array}$} & \BlUE{$\begin{array}{r@{~}c@{~}l} \m{b} & \to & \m{a} \end{array}$} \onslide<4-> \\[.5ex] confluence & \GREEN{$\begin{array}{r@{~}c@{~}l} \m{a} & \to & \m{b} \end{array}$} & \BlUE{$\begin{array}{r@{~}c@{~}l} \m{a} & \to & \m{c} \end{array}$} \end{tabular} \] \end{remark} \medskip \begin{definition}<5-> \smallskip A property $P$ is \alert{preserved under signature extension} if: \[ (\FF,\RR) \vDash P \quad\Longrightarrow\quad (\FF \cup \GG,\RR) \vDash P \] for all TRSs $(\FF,\RR)$ and signatures $\GG$ with $\FF \cap \GG \neq \varnothing$. \smallskip \end{definition} \end{frame}