Multiplicative structure of biorthomorphisms and embedding of orthomorphisms

Let XX be an Archimedean vector lattice. A biorthomorphism on XX is a bilinear map from X×XX×X into XX which is an orthomorphism on XX in each variable separately. The set of such biorthomorphisms is denoted by Orth(X,X). We prove that if Orth(X,X) is not trivial then Orth(X,X) is equipped with a structure of ff-algebra, giving thus a complete answer to a question asked quite recently by Buskes, Page, and Yilmaz. On the other hand, we assume that XX is a semiprime ff-algebra and we show that if XX is either Dedekind-complete or uniformly-complete with a weak order unit, then the set of all orthomorphisms on XX has an order ideal copy in Orth(X,X). Notice that the Dedekind-complete case has been obtained again by Buskes, Page, and Yilmaz in a completely different way.