Exercise 11, § 2

Let $\Sigma$ be a species of structures (in a theory $\mathscr{T}$) which has only one base set. Let $s\in F(x)$ be the typical characterization, and $R\{x,s\}$ the axiom of $\Sigma$. Let $A(x)$ denote the set of structures of species $\Sigma$ on $x$. Let $\sigma\{x,y,s,t\}$ be a term which satisfies the conditions $(\mathrm{MO}_I)$, $(\mathrm{MO}_{II})$, and the following condition :

$(\mathrm{MO}'_{III})$ Given two sets E, E' (in a theory $\mathscr{T}'$ which is stronger than $\mathscr{T}$) endowed with structures $\mathscr{S}$, $\mathscr{S}'$ of species $\Sigma$, respectively, if $f$ is any isomorphism of E onto E', then $f\in\sigma\{E,E',\mathscr{S},\mathscr{S}'\}$.

(a) Let $I_x$ be the identity mapping of $x$ onto itself. Show that the relation $Q\{x,s,t\}$ :

is an equivalence relation between $s$ and $t$ on $A(x)$. Let $B(x)$ be the quotient set $A(x)/Q$, and let $\varphi_x$ be the canonical mapping of $A(x)$ onto $B(x)$. Suppose that the relation $s'\in B(x)$ is transportable (see the Appendix for conditions which will ensure that this is so), and let $\Theta$ denote the species of structures with typical characterization $s'\in \mathfrak{P}(F(x))$ and axiom $s'\in B(x)$.

(b) Let $\bar{\sigma}\{x,y,s',t'\}$ be the set of mappings $f\in\mathscr{T}(x,y)$ which satisfy the following relation : “$s'\in B(x)$ and $t'\in B(y)$ and there exist $s\in A(x)$ and $t\in A(y)$ such that $s'=\varphi_x(s)$ and $t'=\varphi_y(t)$ and $f\in\sigma\{x,y,s,t\}$.” Show that, for the species of structures $\Theta$, the term $\bar{\sigma}$ satisfies $(\mathrm{MO}_I)$, $(\mathrm{MO}_{II})$, and $(\mathrm{MO}_{III})$ and that we have