Exercise 16, § 2

Let $\tau$ be an associative law on a set $E$ and $E^*$ the set of cancellable elements of $E$; suppose that $E^*$ is non-empty and that every cancellable element is a central element. Let $\mathscr{F}$ denote the set of subsets $X$ of $E$ with the following property: there exists $y \in E^*$ such that $\delta_y(E) \subset X$.

(a) Show that the intersection of two sets of $\mathscr{F}$ belong to $\mathscr{F}$.

(b) Let $\Phi$ be the set of functions $f$ defined on a set of $\mathscr{F}$ taking their values in $E$ and such that, for $X \in \mathscr{F}$, $f^{-1}(X)$ belongs to $\mathscr{F}$. Let $R$ denote the relation between elements $f$ and $g$ of $\Phi$: "there exists a set $X \in \mathscr{F}$ such that the restrictions of $f$ and $g$ to $X$ are identical". Show that $R$ is an equivalence relation; let $\Psi = \Phi / R$ be the quotient set of $\Phi$ under this relation.

(c) Let $f$ and $g$ be two elements of $\Phi$, $A \in \mathscr{F}$ and $B \in \mathscr{F}$ the sets where $f$ and $g$ are respectively defined; there exists $X \subset B$ and belonging to $\mathscr{F}$ such that $g(X) \subset A$; if $g_X$ is the restriction of $g$ to $X$, show that the mapping $f \circ g_X$ belongs to $\Phi$ and that its class (mod. $R$) depends only on the classes of $f$ and $g$ (and not on $X$); this class is called the composition of that of $f$ and that of $g$; show that the law of composition thus defined on $\Psi$ is associative and has an identity element.

(d) For all $a \in E$, show that the left translation $\gamma_a$ belongs to $\Phi$; let $\phi_a$ be its class (mod. $R$). Show that the mapping $x \mapsto \phi_x$ is an isomorphism of $E$ onto a submonoid of $\Psi$ and that, if $x \in E^*$, $\phi_x$ is invertible in $\Psi$ (consider the inverse mapping of $\gamma_x$, show that it belongs to $\Phi$ and that its class (mod. $R$) is the inverse of $\phi_x$). Deduce a generalization of Theorem 1 of no. 4.