Constrained supremum and supremum operators are introduced to obtain a general procedure for computing supremal elements of upper semilattices. Examples of such elements include supremal (A; B)-invariant subspaces in linear system theory and supremal controllable sublanguages in discrete-event system theory. For some examples, we show that the algorithms available in the literature are special cases of our procedure. Our iterative algorithms may also provide more insight into applications; in the case of supremal controllable subpredicate, the algorithm enables us to derive a lookahead policy for supervisory control of discrete-event systems.