Exercise 1, § 1

Let $S$ be the set of signs $P$, $X$, $x_1,\ldots,x_n$, the letters $x_i$ being of weight $0$, $P$ of weight $1$, and $X$ of weight $2$. Let $T$ be a balanced word in $L(S)$. Such a word will be called an echelon type on $x_1,\ldots,x_n$.

Let $E_1,\ldots,E_n$ be $n$ terms in a theory stronger than the theory of sets. For each echelon type $T$ on $x_1,\ldots,x_n$ define a term $T(E_1,\ldots,E_n)$ as follows :

(i) If $T$ is a letter $x_i$, then $T(E_1,\ldots,E_n)$ is the set $E_i$.

(ii) If $T$ is of the form $PU$, then $T(E_1,\ldots,E_n)$ is the set

(iii) If $T$ is of the form $XUV$, where $U$ and $V$ are assemblies antecedent to $T$, then $T(E_1,\ldots,E_n)$ is the set $U(E_1,\ldots,E_n)\times V(E_1,\ldots,E_n)$.

Show that, for each echelon type $T$ on $x_1,\ldots,x_n$, $T(E_1,\ldots,E_n)$ is an echelon on the terms $E_1,\ldots,E_n$, and conversely (proof by induction on the length of the echelon type or of the echelon construction scheme). $T(E_1,\ldots,E_n)$ is called the realization of the echelon type $T$ on the terms $E_1,\ldots,E_n$. More precisely, every echelon on $n$ distinct letters can be written in exactly one way in the form $T(x_1,\ldots,x_n)$, where $T$ is an echelon type.

Also show how one may associate with an echelon type $T$ on $n$ letters, and $n$ mappings $f_1,\ldots,f_n$, a canonical extension of these mappings. Deduce that if two echelon construction schemes $S$, $S'$ on $n$ letters are such that $S(x_1,\ldots,x_n)=S'(x_1,\ldots,x_n)$ ($x_1,\ldots,x_n$ being distinct letters), then