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main.tex

@@ -115,20 +115,6 @@ We call string-to-string (monadic) interpretations the particular case when the
 
 An \expreg function is a string-to-string monadic interpretation.
 We denote by \textsf{ExpF} the class of \expreg functions.
-As an example, consider the following function. Let $\Sigma=\{a,b\}$
-and $\Gamma = \Sigma{\times} \{0,1\}\cup \set{\#}$. Let $u\in\Sigma^*$.
-Given a
-subset $U\subseteq Pos(u)$, we let $u_U\in \Gamma^*$ such that
-$|u_U| = |u|$ and for all positions $p$, $u_U(p) = (u(p), p\in
-U)$. Given another subset $V$, $U\leq_{lex} V$ if $U = V$ or the
-smallest $x$ such that $x\in V$ iff $x\not\in U$ satisfies $x\in V$
-and $x\not\in U$. Finally, we let $subsets(u) = \prod_{U\subseteq Pos(u) \text{ in
-    lexicographic order}} u_U\#$. For example,
-$$
-subsets(ab) = (a,0)(b,0)\#(a,0)(b,1)\#(a,1)(b,0)\#(a,1)(b,1)\#
-$$
-It is easy to see that $subsets\in\textsf{ExpF}$. 
-
 
 
 
@@ -252,7 +238,7 @@ pebble transducer model. candidate:
 \end{remark}
 
 \begin{example}
-Let $A=\set{a,b}$. Let us define an \expreg function $f:\left(A\cup \set{\sharp}\right)^*\rightarrow (A\times \set{0,1}\cup \set{\natural})^*$.
+Let $A=\set{a,b}$. Let us define an \expreg function $f:\left(A\cup \set{\sharp}\right)^*\rightarrow (A\times \cup{0,1}\cup \set{\natural})^*$.
 What the function $f$ does is list all subsets of positions (without $\sharp$) and output the word labelled with the corresponding subset. The labelled words are output in a specific order, and separated by $\natural$. Let us decribe the order in which these subsets of positions are listed.
 
 Over a word $u$ without $\sharp$, $f$ outputs the list of all annotations, in the lexicographic order over the subset of positions of $u$, \eg
@@ -292,7 +278,7 @@ The \emph{initial configuration} of $\auta$ over $v$ is the word $(q_0,1)$. Give
 
 \begin{question}
 
-Does $\ipt=\mt\circ\msot$ hold?
+Does $\ipt\subseteq \mt\circ\msot$ hold?
 \end{question}
 \section{Some remaining questions}