tuples

preface.tex

@@ -3,7 +3,7 @@
 \begin{document}
 \chapter*{Preface}
 \addcontentsline{toc}{chapter}{Preface}
-\vskip-6ex
+\vskip-2ex
 \begin{minipage}{0.5\textwidth}\sl
 This book was written to answer one question ``Does a recursion theorist dare to write a book on model theory?''
 
@@ -63,6 +63,6 @@ \chapter*{Preface}
 \item \hyperref[newelski]{Newelski's Theorem\/} on the diameter of Lascar types is proved in an elementary self-contained way.
 \item  \hyperref[external]{Stability and NIP\/} are introduced very briefly.
 We only discuss the properties of externally definable sets, which we identify with \textit{approximable sets}.
-\item \noindent\emph{Syntax highlighting\/} is unusual in a {\gr mathematical text} (though it is the norm in the {\mr\TeX\ source} of the very same text). There are good reasons: it is difficult to be coherent; easy to be eye offending. I have not used these as excuse for not trying.
+\item \noindent\emph{Syntax highlighting\/} looks strange in a text of math. There are good reasons: it is difficult to be {\gr coherent}; it is easy to be {\mr mot}{\gr ely}. These notes may contain a few examples. 
 \end{itemize}
 \end{document}

preliminaries.tex

@@ -67,22 +67,40 @@ \section{Structures}
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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Tuples}
-A \emph{tuple of elements of $A$\/}\index{tuple} is a function $a:\alpha\to A$ from an ordinal $\alpha$ to a set $A$.  We call $\alpha$ the \emph{length} of $a$ and denote it by \emph{$|a|$}\index{length!of a tuple}\index{0a@$|a|$}. If $a$ is surjective, it is said to be an \emph{enumeration\/}\index{enumeration} of $A$. Occasionally, we may say \emph{sequence\/} for tuple --~a word that is also used when the domain is not necessarily an ordinal.
 
-We write \emph{$a_i$} for the $i$-th element of $a$, that is, the element $a(i)$, where $i<\alpha$. We may use the notation \emph{$a=\<a_i: i<\alpha\>$}\index{0a0al@$\<a_i: i<\alpha\>$} for a tuple; when $\alpha$ is finite, we may also write \emph{$a=a_0,\dots,a_{\alpha-1}$}. We write \emph{$a_{\restriction n}$} for the restriction of $a$ to $n$. 
+A \emph{sequence\/} is a function  $a:I\to A$ whose domain is a linear order $I,<_I$.
+We may use the notation \emph{$a=\<a_i: i\in I\>$} for sequences.
+A \emph{tuple\/} is a sequence whose domain is an ordinal, say $\alpha$, then we write  $a=\<a_i: i<\alpha\>$.
+When $\alpha$ is finite, we may also write \emph{$a=a_0,\dots,a_{\alpha-1}$}
+The domain of the tuple $a$, the ordinal $\alpha$, is denoted by \emph{$|a|$\/} and is called the \emph{length\/} of $a$.
+If $a$ is surjective, it is said to be an \emph{enumeration\/}\index{enumeration} of $A$.
 
-\noindent\llap{\textcolor{red}{\Large\danger}\kern1.5ex}In general, note that $I\subseteq\alpha$ has a unique increasing enumeration, say $\<i_i:k<\beta\>$. Then we write \emph{$a_{\restriction I}$} for the tuple $\<a_{i_k}: k<\beta\>$.
+If $J\subseteq I$ is a subset of the domain of the sequence $a=\<a_i: i\in I\>$, we write  \emph{$a_{\restriction J}$} for the restriction of $a$ to $J$.
+When $J$ is well ordered by $<_I$, e.g.\@ when $a$ is a tuple or when $J$ is finite, we identify $a_{\restriction J}$ with a tuple.
+This is the tuple $\<a_{j_k}: k<\beta\>$ where $\<j_k:k<\beta\>$ is the unique increasing enumeration on $J$.
 
-\noindent\llap{\textcolor{red}{\Large\danger}\kern1.5ex}We sometimes overline %overline non esiste come verbo
- the symbol that denotes a tuple. For ease of notation, the elements of the tuple \emph{$\bar a$\/} are denoted by $a_i$ (without the bar). Similarly for $a_{\restriction n}$ and $a_{\restriction I}$.
+\noindent\llap{\textcolor{red}{\Large\danger}\kern1.5ex}Sometimes (i.e.\@ not always) we may overline tuples or sequences as mnemonic.
+When a tuple $\bar c$ is introduced, we write $c_i$ for the $i$-th element of $\bar c$.
+and $c_{\restriction J}$ for the restriction of $\bar c$ to $J\subseteq |\bar c|$.
+Note that the bar is dropped for ease of notation.
 
-The set of tuples of elements of length $\alpha$ is denoted by \emph{$A^\alpha$}. The set of tuples of length $<\alpha$ is denoted by \emph{$A^{<\alpha}$}. For instance, \emph{$A^{<\omega}$\/} is the set of all finite tuples of elements of $A$. When $\alpha$ is finite we do not distinguish between $A^\alpha$ and the $\alpha$-th Cartesian power of $A$. In particular, we do not distinguish between $A^1$ and $A$.
+The set of tuples of elements of length $\alpha$ is denoted by \emph{$A^\alpha$}.
+The set of tuples of length $<\alpha$ is denoted by \emph{$A^{<\alpha}$}.
+For instance, \emph{$A^{<\omega}$\/} is the set of all finite tuples of elements of $A$.
+When $\alpha$ is finite we do not distinguish between $A^\alpha$ and the $\alpha$-th Cartesian power of $A$.
+In particular, we do not distinguish between $A^1$ and $A$.
 
-If $a,b\in A^{|\alpha|}$ and $h$ is a function defined on $A$, we write $h(a)=b$ for $h(a_i)=b_i$. We often do not distinguish between the pair $\<a,b\>$ and the tuple of pairs $\<a_i,b_i\>$. The context will resolve the ambiguity.
+If $a,b\in A^{\alpha}$ and $h$ is a function defined on $A$, we write $h(a)=b$ for $h(a_i)=b_i$.
+We often do not distinguish between the pair $\<a,b\>$ and the tuple of pairs $\<a_i,b_i\>$.
+The context will resolve the ambiguity.
 
-Note that there is a unique tuple of length $0$, the empty set $\0$, which in this context is called \emph{empty tuple\/}\index{empty tuple}. Recall that by definition \emph{$A^{0}$\/}$=\{\0\}$ for every set $A$. Therefore, even when $A$ is empty, $A^{0}$ contains the empty string. 
+Note that there is a unique tuple of length $0$, the empty set $\0$, which in this context is called \emph{empty tuple\/}\index{empty tuple}.
+Recall that by definition \emph{$A^{0}$\/}$=\{\0\}$ for every set $A$.
+Therefore, even when $A$ is empty, $A^{0}$ contains the empty string.
 
-We often concatenate tuples. If $a$ and $b$ are tuples, we write \emph{$a\,b$\/} or, equivalently,\ \emph{$a,b$\/}. 
+
+We often concatenate tuples.
+If $a$ and $b$ are tuples, we write \emph{$a\,b$\/} or, equivalently,\ \emph{$a,b$\/}.