Exercise 10, § 3

Let $F$ be a set, $C$ a subset of $F\times F$ and $m: C\rightarrow F$ a mapping. We make the following hypotheses:

(i) The mappings from C into $F\times F$ given by $(f, g)\mapsto (f, m(f, g))$ and $(f, g)\mapsto (m(f, g), g)$ are injective ;

(ii) If $f,g,h\in F$ are such that the pairs $(f, g)$ and $(g, h)$ belong to C, then the pairs $(f, m(g, h))$ and $(m(f, g), h)$ belong to C and one has $m(f, m(g, h)) =m(m(f, g), h)$ ;

(iii) For every $f\in F$, there exist elements $o(f),t(f)$ and $\overline{f}\in F$, necessarily unique by virtue of (i), such that the pairs $(f, t(f))$, $(o(f), f)$, $(f, f)$, $(f , f)$ belong to C and such that one has $m(f, t(f)) =f=m(o(f), f)$ and $m(f, f) =o(f)$.

a) Prove that, for every $f\in F$, one has $(f , f)\in C$ and $m(f , f) =t(f)$. Deduce that$o(\overline{f}) =t(f)$ and$(o(\overline{\overline{f}}f=), of(.f))\in C$ and $m(o(f), o(f)) =o(f)$. Also prove that

b) Let S be the set of elements $e$ of F such that $(e, e)\in C$ and $m(e, e) =e$; let $o$ and $t$ denote the mappings from F into S given by $f\mapsto o(f)$ and $f\mapsto t(f)$.

Prove that the quadruplet $\Gamma = (S,F, o, t)$ is a quiver and that the mapping $m$ is a composition law in Γ which makes it a groupoid. (This construction is due to H. Brandt ; cf. « Über eine Verallgemeinerung des Gruppenbegriffes », Mathematische Annalen (1926), vol. 96, p. 360–366.)