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We study the poset NO(B) of necessity operators on a boolean algebra B. Weshow 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 isatomic. In that case, NO(B) is a locally Stone frame. Dual results hold for theposet PO(B) of possibility operators. We also obtain similar results for theposets TNO(B) and TPO(B) of tense necessity and possibility operators on B. Ourmain tool is Jonsson-Tarski duality, by which such operators correspond tocontinuous and interior relations on the Stone space of B.

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