\subsection{Relational structures and logical interpretations}
\subsubsection{Relational structures}
\paragraph{Signature}
A \emph{signature}$\ssign$ is a set $S$ of \emph{symbols}\footnote{We only consider relational signatures.}, together with an \emph{arity function}, which we denote by $\ar:S\rightarrow\nat$. Abusing notations, we will often write $R\in\ssign$ instead of $R\in S$.
\paragraph{Structures} A \emph{structure}$w$ over a signature $\ssign$ is given as a \emph{domain}$D$ together with a function which maps any symbol $R$ of $\ssign$ to a set denoted $R^w$ such that $R^w\subseteq D^r$, with $r=\ar(R)$.
\paragraph{Operations} The \emph{disjoint union} of two structures over
\paragraph{The orderned model of words}
\paragraph{The powerset model of words}~\\
Any \mso-formula can be seen as an \fo-formula over the powerset model
Any \mso-formula can be seen as an \fo-formula over the powerset model.
Given a structure $A$ we denote the powerset of $A$ the structure $\pow A$.