Multimodal logics of products of topologies
If modal logics \(L_1,L_2\) with modalities \(\square_1,\square_2\) are determined by classes \(\mathbb{F}_1,\mathbb{F}_{2}\) of Kripke frames, then \(L_1 \times L_2\) is determined by the class of products \(\mathbb{F}_1 \times \mathbb{F}_{2}= \langle W_1 \times W_2,R_1,R_2\rangle\) and is axiomatized (by D. Gabbay and V. Shekhtman, under suitable conditions) by Fusion \(L_1+L_2\) plus \(com=\square_1\square_2p\to\square_2\square_1p\) and \(chr=\diamondsuit_1\square_2p\to\square_2\diamondsuit_1p\). In the topological semantics, when \(\square\) is interpreted as the interior of a set, products \(X\times Y\) not always validate \(com\) and \(chr\). The authors prove that both \(X\) and \(Y\) being Alexandrov spaces (intersection of an arbitrary family of open sets is open) is sufficient. However, for rationals \(\mathbb{Q}\), the logic of \(\mathbb{Q}\times \mathbb{Q}\) is complete for the fusion \(\text{S}4+\text{S}4\), hence much weaker than \(\text{S}4 \times \text{S}4\). A new completeness proof of S4 for \(\mathbb{Q}\) is presented. The authors introduce and investigate new kinds of topologies, horizontal and vertical, which they call coordinate topologies. In the typical case of \(\mathbb{R}\times \mathbb{R}\) a set \(A\) is horizontally open if it contains with each point \((x,y)\) a horizontal interval \((a,b)\times \{y\}\) for \(a<x<b\) and similarly for the vertical topology.
scientific article; zbMATH DE number 5119613