Exercise 2, § 1

Let E be a lattice with a composition law (x, y) $\mathscr{H} xy$ (not necessarily associative), such that, for all a ∈ E, the maps x $\mathscr{H} ax$ and x $\mapsto xa$ are order-automorphisms of E. Let $x_a$ (resp. $_a x$) denote the element of E defined by $(x_a)a = x$ (resp. $a(_a x) = x$) and suppose that for each a ∈ E the maps x $\mathscr{H} a_x$ and x $\mathscr{H} _x a$ are isomorphisms from the ordering of E to the opposite ordering. Show that, for arbitrary x, y, z,