diff --git a/main.tex b/main.tex
index 02d256b5f07090fc0618bcf7ed4bb8529e2fb6ff..32fbf416cf080c3cd37772dad77789283407034f 100644
--- a/main.tex
+++ b/main.tex
@@ -8,6 +8,7 @@
 \newcommand{\Expreg}{Exp-regular\xspace}
 
 
+
 \newcommand{\mso}{\textsf{MSO}\xspace}
 \newcommand{\fo}{\textsf{FO}\xspace}
 \newcommand{\msot}{\textsf{MSO-T}\xspace}
@@ -17,7 +18,8 @@
 \newcommand{\msomi}{\textsf{MSO-MI}\xspace}
 \newcommand{\fomi}{\textsf{FO-MI}\xspace}
 \newcommand{\nmsomi}{\textsf{NMSO-MI}\xspace}
-
+\newcommand{\pow}[1]{\mathsf{2}^{#1}}
+\newcommand{\ar}{\mathsf{ar}}
 
 \newcommand{\sst}{\textsf{SST}\xspace}
 \newcommand{\mt}{\textsf{MT}\xspace}
@@ -37,6 +39,8 @@
 \newcommand{\tuple}[1]{\left(#1\right)}
 \newcommand{\sem}[1]{\left\llbracket #1\right\rrbracket}
 
+\newcommand{\nat}{\mathbb{N}}
+
 \newcommand{\eg}{\textit{e.g.~}}
 \newcommand{\ie}{\textit{i.e.~}}
 
@@ -84,11 +88,20 @@
 \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 $\pow A$.
 \subsubsection{Logical interpretation}
 Here define \msoi and \foi.
 \subsection{Monadic interpretation}