On Modal Logics of Full Products of Neighborhood Frames

Abstract

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.