On Modal Logics of Full Products of Neighborhood Frames

On the product of two neighborhood frames, three natural neighborhood functions can be defined: the horizontal one assigning to a point (x, y) the set of all supersets of sets U × y, where U is a neighborhood of x; the vertical analog; and the product neighborhood function assigning as neighborhoods all supersets of sets U ×V, for neighborhoods U of x and V of y. We define the tri-modal logics T × + n T and D × + n D of classes of full products equipped with all three neighborhood functions of neighborhood frames validating the logic T or D; thereby extending known product results for S4 and D4 to weaker systems. Two interaction principles arise: where □ for the product neighborhood function and □ 1, □ 2 the horizontal and vertical ones. Namely, we show that T × + n T = T ⊗ T ⊗ T + (mix) and D × + n D = D ⊗ D ⊗ D + (mix), where ⊗ denotes fusion. Notably, (sub) and (mix) are equivalent over S4 ⊗ S4 ⊗ S4 and thus S4 ⊗ S4 ⊗ S4 + (mix) axiomatizes the logic of full products of topological spaces.