A note on weak orthomorphisms (Q1024150)

Let \(E\) be a relatively uniformly complete vector lattice, \(D\) a vector sublattice. An order-bounded linear operator \(T: D\to E\) is called an orthomorphism if \(|x|\wedge|y|= 0\) in \(D\) implies \(|T(x)|\wedge|y|= 0\) in \(E\). It is weak if \(D\) is order-dense in \(E\). A weak orthomorphism is called extended orthomorphism if \(D\) is an order ideal in \(E\). The author proves that weak orthomorphism and extended orthomorphism coincide.