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THE MCKINSEY–TARSKI THEOREM FOR LOCALLY COMPACT ORDERED SPACES

Part of: Peculiar spaces Fairly general properties Special properties General logic

Published online by Cambridge University Press:  29 April 2021

GURAM BEZHANISHVILI ,
NICK BEZHANISHVILI ,
JOEL LUCERO-BRYAN  and
JAN VAN MILL
GURAM BEZHANISHVILI
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES NEW MEXICO STATE UNIVERSITYLAS CRUCES, NM, USAE-mail: guram@nmsu.edu
NICK BEZHANISHVILI
Affiliation:
INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAMAMSTERDAM, THE NETHERLANDSE-mail: N.Bezhanishvili@uva.nl
JOEL LUCERO-BRYAN
Affiliation:
DEPARTMENT OF MATHEMATICS KHALIFA UNIVERSITY OF SCIENCE AND TECHNOLOGY ABU DHABI, UNITED ARAB EMIRATESE-mail: joel.bryan@ku.ac.ae
JAN VAN MILL
Affiliation:
KORTEWEG-DE VRIES INSTITUTE FOR MATHEMATICS UNIVERSITY OF AMSTERDAMAMSTERDAM, THE NETHERLANDSE-mail: j.vanMill@uva.nl
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Abstract

We prove that the modal logic of a crowded locally compact generalized ordered space is $\textsf {S4}$ . This provides a version of the McKinsey–Tarski theorem for generalized ordered spaces. We then utilize this theorem to axiomatize the modal logic of an arbitrary locally compact generalized ordered space.

Keywords

modal logictopological semanticslinearly ordered spacegeneralized ordered spacelocally compact spacescattered space

MSC classification

Primary: 03B45: Modal logic (including the logic of norms)
Secondary: 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 54D45: Local compactness, $sigma$-compactness 54G12: Scattered spaces 03B55: Intermediate logics
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 27 , Issue 2 , June 2021 , pp. 187 - 211
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Copyright
© The Association for Symbolic Logic 2021

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