On extensions of some nonlinear maps in vector lattices (Q2013095)

An operator \(T:E\to F\) (acting on vector lattices\()\) is said to be an abstract Urysohn operator if it is orthogonally additive (i.e., \( T(x+y)=Tx+Ty\) for all \(x,y\in E\), \(x\bot y)\), and maps order bounded sets to order bounded sets; \(T\) is said to be an Urysohn lattice homomorphism if, in addition, \(T(x\vee y)=Tx\vee Ty\) and \(T(x\wedge y)=Tx\wedge Ty\) for all \( x,y\in E_{+}.\) Theorem 2.8 indicates very general conditions under which an even operator \(T\) defined on a normal sublattice \(D\) of \(E\) and taking values in \(F_{+}\) admits a factorization of the form \(T=j\circ \tilde{T} \circ i,\) where \(i:D\to E\) is the canonical inclusion, \(\tilde{T} :E\to F\) is an Urysohn lattice homomorphism, and \(j:F\to F_{+} \) is the map defined by \(j(x)=x_{+}\). Another important result is Theorem 3.11, which clarifies the structure of positive orthogonally additive operators dominated by an Urysohn lattice homomorphism defined on a vector lattice \(E\) and taking value in a Dedekind complete vector lattice \(F\).