On Disjointness-Preserving Biadditive Operators

Orthogonally biadditive operators preserving disjointness are studied. It is proved that, that for a Dedekind complete vector lattice $W$ and order ideals $E$ and $F$ in $W$, the set $\mathfrak{N}(E,F;W)$ of all orthogonally biadditive operators commuting with projections is a band in the Dedekind complete vector lattice $\mathcal{OBA}_r(E,F;W)$ of all regular orthogonally biadditive operators from the Cartesian product of $E$ and $F$ to $W$. A general form of the order projection onto this band is found, and an operator version of the Radon–Nikodym theorem for disjointness-preserving positive orthogonally biadditive operators is proved.