Exercise 22, § 4

Let E be a quasi-group (§ 3, Exercise 6) written multiplicatively, possessing an idempotent e and such that $(xy)(zt) = (xz)(yt)$ for all $x, y, z, t$. Let $u(x)$ denote the element of E such that $u(x)e = x$ and $v(x)$ the element of E such that $ev(x) = x$; show that the law of composition $(x, y) \mapsto u(x)v(y)$ is a commutative group law on E under which $e$ is identity element, that the mappings $x \mapsto xe$ and $y \mapsto ey$ are permutable endomorphisms of this group structure and that $xy = u(xe)v(ey)$ (start by establishing the identities $e(xy) = (ex)(ey)$, $(xy)e = (xe)(ye)$, $e(xe) = (ex)e$; then note that the relations $x = y$, $ex = ey$ and $xe = ye$ are equivalent). Obtain the converse.