Orthomorphisms on non-Banach F-lattices containing a copy of RN with applications to Musielak-Orlicz spaces

Let E=(E,‖⋅‖) be a (non-Banach) F-lattice with the σ-Fatou property. We prove that if E contains a subspace linearly homeomorphic to the F lattice ω=RN then the lattice Orth(E) - of all orthomorphisms on E - contains a linear-lattice copy of ω. In particular, Orth(E) contains noncentral orthomorphisms. In the case of Musielak-Orlicz F-lattices we show that this property can be expressed by means of a property of Musielak-Orlicz functions.