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KRULL DIMENSION IN MODAL LOGIC

Published online by Cambridge University Press:  09 January 2018

GURAM BEZHANISHVILI ,
NICK BEZHANISHVILI ,
JOEL LUCERO-BRYAN  and
JAN VAN MILL
GURAM BEZHANISHVILI
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES NEW MEXICO STATE UNIVERSITY LAS CRUCES, NM88003, USAE-mail: guram@math.nmsu.edu
NICK BEZHANISHVILI
Affiliation:
INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM1090GE AMSTERDAM, THE NETHERLANDSE-mail: n.bezhanishvili@uva.nl
JOEL LUCERO-BRYAN
Affiliation:
DEPARTMENT OF APPLIED MATHEMATICS AND SCIENCES KHALIFA UNIVERSITY PO127788ABU DHABI, UAEE-mail: joel.lucero-bryan@kustar.ac.ae
JAN VAN MILL
Affiliation:
KORTEWEG-DE VRIES INSTITUTE FOR MATHEMATICS UNIVERSITY OF AMSTERDAM1098XG AMSTERDAM, THE NETHERLANDSE-mail: j.vanmill@uva.nl
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Abstract

We develop the theory of Krull dimension for S4-algebras and Heyting algebras. This leads to the concept of modal Krull dimension for topological spaces. We compare modal Krull dimension to other well-known dimension functions, and show that it can detect differences between topological spaces that Krull dimension is unable to detect. We prove that for a T1-space to have a finite modal Krull dimension can be described by an appropriate generalization of the well-known concept of a nodec space. This, in turn, can be described by modal formulas zemn which generalize the well-known Zeman formula zem. We show that the modal logic S4.Zn := S4 + zemn is the basic modal logic of T1-spaces of modal Krull dimension ≤ n, and we construct a countable dense-in-itself ω-resolvable Tychonoff space Zn of modal Krull dimension n such that S4.Zn is complete with respect to Zn. This yields a version of the McKinsey-Tarski theorem for S4.Zn. We also show that no logic in the interval [S4n+1S4.Zn) is complete with respect to any class of T1-spaces.

Keywords

03B4506E2506D2054F4554B17Krull dimensionmodal logictopological semanticsHeyting algebra
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 4 , December 2017 , pp. 1356 - 1386
Copyright
Copyright © The Association for Symbolic Logic 2017 

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