On the structure of modal and tense operators on a boolean algebra

We initiate the study of the poset \(\mathcal{N}\mathcal{O}(B)\) of necessity operators on a boolean algebra B. We show that \(\mathcal{N}\mathcal{O}(B)\) is a meet-semilattice that need not be distributive. However, when B is complete, \(\mathcal{N}\mathcal{O}(B)\) is necessarily a frame, which is spatial iff B is atomic. In that case, \(\mathcal{N}\mathcal{O}(B)\) is a locally Stone frame. Dual results hold for the poset \(\mathcal{P}\mathcal{O}(B)\) of possibility operators. We also obtain similar results for the posets \(\mathcal {TNO}(B)\) and \(\mathcal {TPO}(B)\) of tense necessity and possibility operators on B. Our main tool is Jónsson-Tarski duality, by which such operators correspond to continuous and interior relations on the Stone space of B.