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Abstract. Let E be an Archimedean vector lattice, let E d be its Dedekind completion and let B be a Dedekind complete vector lattice. If Ψ 0 : E × E → B is an orthosymmetric lattice bimorphism, then there exists a lattice bimorphism Ψ :that not just extends Ψ 0 but also has to be orthosymmetric. As an application, we prove the following: Let A be an Archimedean d-algebra. Then the multiplication in A can be extended to a multiplication in A d , the Dedekind completion of A, in such a fashion that A d is again a d-algebra with respect to this extended multiplication. This gives a positive answer to the problem posed by C. B. Huijsmans in 1990.

Let A, B and C be lattice-ordered algebras and let Ψ : A × B → C be a bilinear map. We call Ψ a lattice bimorphism if for each 0 ≤ f ∈ A and each 0 ≤ g ∈ B, the partial maps g → Ψ (f, g) and f → Ψ (f, g) are lattice homomorphisms of B and A into C, respectively; and we say that Ψ is multiplicative if ΨIn this paper, we study the connection between lattice bimorphisms and multiplicative bilinear maps on f -algebras in great detail. Our central result in this direction is the following: if A, B and C are Archimedean f -algebras with unit elements e A , e B and e C respectively and Ψ : A × B → C is a Markov bilinear map (i.e., Ψ is positive and Ψ (e A , e B ) = e C ) then Ψ is a lattice bimorphism if and only if Ψ is multiplicative. The main application of this result we present in this work is the Cauchy-Shwarz inequality in Archimedean (not necessarily commutative) d-algebras, which is an improvement of the result of Buskes and van Rooij, who established this inequality in the commutative case.

In this paper, we present some results concerning the automatic order boundedness of band preserving operators on Dedekind σ -complete vector lattices.