\section{Classes of riegs}
\memo{Consider equations $0^x=0, 0^x=1$}
\subsection{Canonically ordered riegs}
\subsection{Strict initial riegs}
\subsection{Locally connected riegs: Connected objects and connected elements}
\begin{definition}
    An element $x\in R$ of a rieg $R$ is said to be \emph{connected} if it satisfies the following equivalent conditions:
    \begin{enumerate}
        \item $(-)^{x}\colon R \to R$ is a rig homomorphism.
        \item $0^x= 0$ and $(a+b)^x=a^x + b^x$ .
        \item ?
    \end{enumerate}
\end{definition}

\begin{proposition}
    For a locally connected rieg, 
    \[0^x = \]
\end{proposition}
\memo{Heyting, locally connected, strict initial}