Disjointness and order projections in the vector lattices of abstract Uryson operators (Q331752)

Let \(E\) and \(F\) be vector lattices. A mapping \(T:E \rightarrow F\) is called orthogonally additive if \(T(x+y)=T(x) + T(y)\) for each \(x,y \in E\) for which \(|x| \land |y|=0\). If \(T\) maps bounded sets of \(E\) to bounded sets of \(F\), then \(T\) is referred to as an \textit{order bounded} map. An orthogonally additive order bounded map is called an abstract Urysohn operator. The map \(T:E \rightarrow F\) is called positive if it maps positive elements of \(E\) to positive elements of \(F\). Let \(F\) be a Dedekind complete vector lattice. Then \({\mathcal U}(E,F)\), the set of all abstract Urysohn operators from \(E\) to \(F\), is a Dedekind complete vector lattice. The authors obtain a characterization for the disjointness of two positive operators in \({\mathcal U}(E,F)\), extending a result known for linear operators, as well as formulae to calculate projections onto a band generated by an increasing set. Certain special cases are presented as corollaries.