Multiplicative structure of biorthomorphisms and embedding of orthomorphisms

Abstract

Let X be an Archimedean vector lattice. A biorthomorphism on X is a bilinear map from X×X into X which is an orthomorphism on X 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 f-algebra, giving thus a complete answer to a question asked quite recently by Buskes, Page, and Yilmaz. On the other hand, we assume that X is a semiprime f-algebra and we show that if X is either Dedekind-complete or uniformly-complete with a weak order unit, then the set of all orthomorphisms on X 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.