The r-asymptotically quasi finite rank operators on Banach lattices are examples of regular Riesz operators. We characterise Riesz elements in a subalgebra of a Banach algebra in terms of Riesz elements in the Banach algebra. This enables us to characterize r-asymptotically quasi finite rank operators in terms of adjoint operators. The r-asymptotically quasi finite rank operators are also employed to study the following problem: Suppose operatorsSandTon a Banach latticeEsatisfy 0 ≤S≤T. IfTis a Riesz operator, when is it true thatSis a Riesz operator?