Finite elements in some vector lattices of nonlinear operators

We study the collection of finite elements Φ 1(U(E, F)) in the vector lattice U(E, F) of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices E and F, where F is Dedekind complete. In particular, for an atomic vector lattice E it is proved that for a finite element in φ∈ U(E, R) there is only a finite set of mutually disjoint atoms, where φ does not vanish and, for an atomless vector lattice the zero-vector is the only finite element in the band of σ-laterally continuous abstract Uryson functionals. We also describe the ideal Φ 1(U(Rn, Rm)) for n, m∈ N and consider rank one operators to be finite elements in U(E, F). © 2017, Springer International Publishing AG.