On extensions of orthosymmetric lattice bimorphisms

Abstract

In the paper we prove that every orthosymmetric lattice bilinear map on the cartesian product of a vector lattice with itself can be extended to an orthosymmetric lattice bilinear map on the cartesian product of the Dedekind completion with itself. The main tool used in our proof is the technique associated with extension to a vector subspace generated by adjoining one element. As an application, we prove that if $(A,\ast )$ is a commutative $d$-algebra and $A^{\mathfrak {d}}$ its Dedekind completion, then, $A^{\mathfrak {d}}$ can be equipped with a $d$-algebra multiplication that extends the multiplication of $A$. \endgraf Moreover, we indicate an error made in the main result of the paper: M. A. Toumi, Extensions of orthosymmetric lattice bimorphisms, Proc. Amer. Math. Soc. 134 (2006), 1615–1621.