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TAMING THE ‘ELSEWHERE’: ON EXPRESSIVITY OF TOPOLOGICAL LANGUAGES

Published online by Cambridge University Press:  28 March 2022

DAVID FERNÁNDEZ-DUQUE*
Affiliation:
INSTITUTE OF COMPUTER SCIENCE OF THE CZECH ACADEMY OF SCIENCES PRAGUE, CZECHIA and DEPARTMENT OF MATHEMATICS WE16 GHENT UNIVERSITYGHENT, BELGIUME-mail: fernandez@cs.cas.cz

Abstract

In topological modal logic, it is well known that the Cantor derivative is more expressive than the topological closure, and the ‘elsewhere’, or ‘difference’, operator is more expressive than the ‘somewhere’ operator. In 2014, Kudinov and Shehtman asked whether the combination of closure and elsewhere becomes strictly more expressive when adding the Cantor derivative. In this paper we give an affirmative answer: in fact, the Cantor derivative alone can define properties of topological spaces not expressible with closure and elsewhere. To prove this, we develop a novel theory of morphisms which preserve formulas with the elsewhere operator.

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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