Lattice effect algebras with (o)-continuous faithful valuations

We prove that if there exists an order-continuous, faithful valuation ω on a lattice effect algebra E then E is modular, separable and order-continuous. It is also shown that such an effect algebra E can be supremum and infimum densely embedded into a complete effect algebra E which is also modular separable and order-continuous, since the valuation ω can be extended to a unique order-continuous faithful valuation ω on E.