The lateral order on Köthe–Bochner spaces and orthogonally additive operators

In this paper, we introduce a new class of regular orthogonally additive operators defined on a lattice-normed space \((\mathcal {X},E)\) and taking values in a vector lattice F. We show that the vector space \(\mathcal{O}\mathcal{A}_r(\mathcal {X},F)\) of all regular orthogonally additive operators from a d-decomposable lattice-normed space \((\mathcal {X},E)\) to a Dedekind complete vector lattice F is a Dedekind complete vector lattice and the lattice operations can be calculated by the Riesz–Kantorovich formulas. We find necessary and sufficient conditions for an orthogonally additive operator \(T:\mathcal {X}\rightarrow F\) to be dominated and obtain a criterion of the positivity of a nonlinear superposition operator \(T_N:E(X)\rightarrow E\) defined on Köthe–Bochner space E(X) and taking values in Köthe-*Banach space E. Finally, we state some open problems.