On modal logics arising from scattered locally compact Hausdorff spaces |
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Authors: | Guram Bezhanishvili Nick Bezhanishvili Joel Lucero-Bryan Jan van Mill |
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Institution: | 1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA;2. Institute for Logic, Language and Computation, University of Amsterdam, 1090 GE Amsterdam, the Netherlands;3. Department of Mathematics, Khalifa University of Science and Technology, PO box 127788 Abu Dhabi, United Arab Emirates;4. Korteweg–de Vries Institute for Mathematics, University of Amsterdam, 1098 XG Amsterdam, the Netherlands |
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Abstract: | For a topological space X, let be the modal logic of X where □ is interpreted as interior (and hence ◇ as closure) in X. It was shown in 3] that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, (), and their intersections arise as for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer to 3, Question 6.2]. On the other hand, we show that a scattered Stone space that is in addition hereditarily paracompact does not give rise to a new logic; namely we show that the logic of such a space is either S4.Grz or for some . In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space. |
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Keywords: | 03B45 03B55 06E15 54D45 54G12 Modal logic Topological semantics Stone space Locally compact space Scattered space Paracompact space |
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