Exercise 23, § 4

Consider on a set E an internal law, not everywhere defined, written multiplicatively and satisfying the following conditions: (1) if one of the compositions $(xy)z, x(yz)$ is defined, so is the other and they are equal; (2) if $x, x', y$ are such that $xy$ and $x'y$ (resp. $yx$ and $yx'$) are defined and equal, then $x = x'$; (3) for all $x \in E$, there exist three elements $e_x, e'_x$ and $x^{-1}$ such that $e_x x = x$, $x e'_x = x$, $x^{-1} x = e'_x$; $e_x$ is called the left unit of $x$, $e'_x$ the right unit of $x$, $x^{-1}$ the inverse of $x$ (by an abuse of language). (a) Show that the compositions $xx^{-1}, x^{-1}e_x, e'_x x^{-1}, e_x e'_x, e'_x e'_x$ are defined and that $xx^{-1} = e_x, x^{-1}e_x = e'_x x^{-1} = x^{-1}, e_x e'_x = e_x, e'_x e'_x = e'_x$. (b) Every idempotent $e$ of E (§ 1, no. 4) is a left unit for all the $x$ such that $ex$ is defined and right unit for all the $y$ such that $ye$ is defined. (c) For the composition $xy$ to be defined, it is necessary and sufficient that the right unit of $x$ be the same as the left unit of $y$ (to see that the condition is sufficient, use the relation $e_y = yy^{-1}$); if $xy = z$, then $x^{-1}z = y, zy^{-1} = x, y^{-1}x^{-1} = z^{-1}, z^{-1}x = y^{-1}, yz^{-1} = x^{-1}$ (the compositions appearing on the left-hand sides of these relations being defined). (d) For any two idempotents $e, e'$ of E, let $G_{e, e'}$ denote the set of $x \in E$ such that $e$ is left unit and $e'$ is right unit of $x$. Show that, if $G_{e, e}$ is non-empty, it is a group under the law induced by that on E. E is called a groupoid if it also satisfies the following condition: (4) for all idempotents $e, e'$, $G_{e, e'}$ is non-empty. (e) In a groupoid E, if $a \in G_{e, e'}$, show that $x \mapsto xa$ is a bijective mapping of $G_{e, e}$ onto $G_{e, e'}, y \mapsto ay$ a bijective mapping of $G_{e', e'}$ onto $G_{e, e'}$ and $x \mapsto a^{-1}xa$ an isomorphism of the group $G_{e, e}$ onto the group $G_{e', e'}$. (f) Show that the law defined in § 1, Exercise 4(b) determines a groupoid structure on the set $\Psi$, if all the sets of the family $\mathscr{F}$ are equipotent.