diff --git a/main.tex b/main.tex
index 9807e302c0a4bbfbcf54ca98f66082cd1d48877e..839e9f4fc4e20c6ddf040f6b8b583a6ce56fd549 100644
--- a/main.tex
+++ b/main.tex
@@ -109,13 +109,15 @@ The \emph{ordered coproduct} of a sequence of structures $A_1,\cdots, A_n$ over
 
 The \emph{ordered product} of a sequence of structures $A_1,\cdots, A_n$ over the same signature $\ssign$ is a structure $\bigotimes_{i\leq n} A_i$ over the signature $\ssign\times\set{\sqsubseteq}$. The domain of $\bigotimes_{i\leq n} A_i$ is $\bigotimes_{i\leq n} \dom_{A_i}$. The interpretation of a symbol $R$ is $\bigoplus_{i\leq n} \inter_{A_i}(R)$. The interpretation of $\sumorder$ is $\bigcup_{i\leq j\leq n}(\dom_{A_i}{\times}\set{i})\times(\dom_{A_j}{\times}\set{j})$.
 
-\paragraph{The orderned model of words}
-
-\paragraph{The powerset model of words}~\\
+The \emph{powerset structure} of a structure $A$ over signature $\ssign$ is a structure $\pow A$ over signature $\ssign\uplus \set{\subseteq}$, where $\subseteq$ is a binary symbol. The domain of $\pow A$ is $\pow{\dom_A}$. The interpretation of a symbol $R$ is the set $\set{(\set{x_1},\dots,\set{x_k})\mid\ (x_1,\ldots,x_n)\in \inter_A(R)}$.
+The interpretation of $\subseteq$ is the actual set inclusion $\set{(x,y)\mid\ x\subseteq y\subseteq \dom_A}$.
+\paragraph{Word models}
+The \emph{ordered model} of a word $w$ over alphabet $\Sigma$ is the usual model with one unary predicate $a$ per letter $a\in \Sigma$, interpreted as the set of positions with letter $a$. Additionally the binary symbol $\leq$ is interpreted as the linear order over the positions of the word.
 
+The \emph{powerset model} of a word is the powerset structure of the structure associated with a word.
 Any \mso-formula can be seen as an \fo-formula over the powerset model.
+We sometimes say \emph{word} to mean its \emph{ordered model} (or even \emph{powerset model}), relying on context to avoid confusion.
 
-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}
@@ -126,22 +128,22 @@ cornercase behaviours with small words.}
 
 Given two relational signatures $\ssign,\tsign$, an \mso monadic interpretation $T$ of dimension $d$ from structures over $\ssign$ to structures over $\tsign$ is given by:
 \begin{itemize}
-\item An \mso-formula $\phi_{U}(\bar X)$ defining the universe of the output structure
+\item An \mso-formula $\phi_{\dom}(\bar X)$ defining the domain of the output structure
 \item For each relation $R\in \tsign$ of arity $k$, a formula $\phi_R(\bar X _1,\ldots, \bar X_k )$
 
 \end{itemize}
 where $\bar X$ denotes a $d$-tuple of monadic variables.
 
-Given a structure $A$ over $\ssign$, we define its image by $T$ by a structure $B=\sem T(A)$ over $\tsign$: the universe of $B$ is the set $U=\set{\bar S\mid\ A \models \phi_U(\bar S)}$, given $R\in \tsign$ of arity $k$, we define $R$ in $B$ as the set $\set{\tuple{\bar S_1,\ldots,\bar S_k}\in U^k\mid\ \phi_R(\bar S_1,\ldots,\bar S_k)}$.
+Given a structure $A$ over $\ssign$, we define its image by $T$ by a structure $B=\sem T(A)$ over $\tsign$: the domain of $B$ is the set $D=\set{\bar S\mid\ A \models \phi_\dom(\bar S)}$, given $R\in \tsign$ of arity $k$, we define $R$ in $B$ as the set $\set{\tuple{\bar S_1,\ldots,\bar S_k}\in \dom_A^k\mid\ \phi_R(\bar S_1,\ldots,\bar S_k)}$.
 %Given tuples $\bar X_1,\ldots,\bar X_l$, we extend $\sem T$ by $\sem T(A,\bar X_1,\ldots,\bar X_l)=B,\set{\bar X_1},\ldots,\set{\bar X_l}$.
 
 
 A non-deterministic \msomi (\nmsomi) $S$ from $\ssign$-structures to $\tsign$-structures with $k$ parameters is given by an \msomi $T$ from $\ssign\uplus\set{X_1,\ldots,X_k}$-structures to $\tsign$-structures where $X_1,\ldots,X_k$ are additional unary symbols. Let $\pi$ denote the natural projection from $\ssign\uplus\set{X_1,\ldots,X_k}$-structures to $\ssign$-structures. We define $\sem S(A)=\set{\sem T(C)\mid\ \pi(C)=A}$.
 
 
-We call string-to-string (monadic) interpretations the particular case when the two signatures have unary symbols and one binary symbol which is a linear order (this can be enforced syntactically).
+We call word-to-word (monadic) interpretations the particular case when the two signatures have unary symbols and one binary symbol which is a linear order (this can be enforced syntactically).
 
-An \expreg function is a string-to-string monadic interpretation.
+An \expreg function is a word-to-word monadic interpretation.
 
 
 
@@ -224,7 +226,7 @@ An \msomi of dimension $d$ has growth $O((2^n)^d)=O(2^{dn})=2^{O(n)}$.
 
 \item \Expreg functions are \emph{not} closed under composition.
 
-\item \Expreg functions are \emph{bot} closed under pre-composition by functions of superlinear growth, in particular polyregular functions.
+\item \Expreg functions are \emph{not} closed under pre-composition by functions of superlinear growth, in particular polyregular functions.
 
 \end{itemize}
 \end{corollary}