Multimodal logics of products of topologies
type: Item
instance of: scholarly article
title: Multimodal logics of products of topologies
zbMATH Open document ID: 1113.03018
author: Guram Bezhanishvili
author: Johan van Benthem
author: Balder ten Cate
author: Darko Sarenac
published in: Studia Logica
publication date: 2007-01-29T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 54B10
zbMATH DE Number: 5119613
zbMATH Keywords: product modalities
zbMATH Keywords: fusion of modal logics
zbMATH Keywords: product of modal logics
zbMATH Keywords: topological product
zbMATH Keywords: horizontal topology
zbMATH Keywords: topological semantics
zbMATH Keywords: vertical topology
reviewed by: G. E. Mints
MaRDI profile type: Publication
full work available at URL: s11225-006-9013-x
OpenAlex ID: W2115647456
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DOI: 10.1007/S11225-006-9013-X
description: scientific article; zbMATH DE number 5119613