  We study the poset NO(B) of necessity operators on a boolean algebra B. We
show that NO(B) is a meet-semilattice that need not be distributive. However,
when B is complete, NO(B) is necessarily a frame, which is spatial iff B is
atomic. In that case, NO(B) is a locally Stone frame. Dual results hold for the
poset PO(B) of possibility operators. We also obtain similar results for the
posets TNO(B) and TPO(B) of tense necessity and possibility operators on B. Our
main tool is Jonsson-Tarski duality, by which such operators correspond to
continuous and interior relations on the Stone space of B.
