\begin{frame} \small \vspace{-1ex} \begin{definitions} \begin{itemize} \item An equation $s \approx t$ is \alert{valid} in the $\Sigma$-algebra $\AA$ \quad(\alert{$\AA \vDash s \approx t$})\quad %, \alert{$s =_\AA t$}) if \[ [s,\alpha]_\AA = [t,\alpha]_\AA \] for all assignments $\alpha$. \item<2-> $\FF$-algebra $\AA$ is \alert{model} of ES $\ES$ if $\AA \vDash s \approx t$ for all equations $s \approx t \in \EE$. \end{itemize} \end{definitions} \smallskip \begin{example}<3-> \begin{itemize} \item $\Aa = ( \NN, [\cdot] )$ with \quad $[0] = 0$, \quad $[s](x) = x+1$, \quad $[+](x,y) = x+y$ \item $\Bb = ( \NN, [\cdot] )$ with \quad $[0] = 1$, \quad $[s](x) = x+1$, \quad $[+](x,y) = 2x+y$ \item ES $\EE$ \[ \GREEN{\begin{array}{r@{~}c@{~}l} \m{0}+y & \approx & y \\[.5ex] \m{s}(x)+y & \approx & \m{s}(x+y) \end{array}} \] \end{itemize} \onslide<4-> \begin{center} $\AA$ is model of $\EE$ \qquad \onslide<5-> $\BB$ is no model of $\EE$ \end{center} \vspace{-2ex} \end{example} \end{frame}