On Modal Logics of Full Products of Neighborhood Frames
For logicians studying modal logics of products, this provides axiomatizations for weaker systems, though the results are incremental extensions of existing work.
The paper defines tri-modal logics for full products of neighborhood frames with three natural neighborhood functions, axiomatizing them as fusions with the interaction principle (mix). It extends known product results for S4 and D4 to weaker systems T and D.
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 the Cartesian product of U and y, where U is a neighborhood of x; the vertical analog; and the product neighborhood function assigning as neighborhoods all supersets of sets Cartesian products of U and V, for neighborhoods U of x and V of y. We define the tri-modal logics Tx+T and Dx+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: (sub) = []p -> [1]p & [2]p and (mix) = []p -> [1][2]p & [2][1]p, where the modality [] stands for the product neighborhood function and [1], [2] the horizontal and vertical ones. Namely, we show that Tx+T = T*T*T + (mix) and Dx+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.