| Affiliation: | 1.Southern Mathematical Institute,Russian Academy of Sciences,Vladikavkaz,Russia;2.RUDN University,Moscow,Russia;3.Department of Mathematics, Institute of Analysis,Technical University Dresden,Dresden,Germany |
| Abstract: | We study the collection of finite elements (Phi _{1}big ({mathcal {U}}(E,F)big )) in the vector lattice ({mathcal {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 (varphi in {mathcal {U}}(E,{mathbb {R}})) there is only a finite set of mutually disjoint atoms, where (varphi ) does not vanish and, for an atomless vector lattice the zero-vector is the only finite element in the band of (sigma )-laterally continuous abstract Uryson functionals. We also describe the ideal (Phi _{1}big ({mathcal {U}}({mathbb {R}}^n,{mathbb {R}}^m)big )) for (n,min {mathbb {N}}) and consider rank one operators to be finite elements in ({mathcal {U}}(E,F)). |
