- Author
- Date
- 12-2017
- Title
- Krull dimension in modal logic
- Journal
- Journal of Symbolic Logic
- Volume | Issue number
- 82 | 4
- Pages (from-to)
- 1356-1786
- Document type
- Article
- Faculty
- Interfacultary Research
Faculty of Science (FNWI) - Institute
- Institute for Logic, Language and Computation (ILLC)
Korteweg-de Vries Institute for Mathematics (KdVI) - 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.
- URL
- go to publisher's site
- Language
- English
- Note
- © The Association for Symbolic Logic 2017
- Persistent Identifier
- https://hdl.handle.net/11245.1/a5a3ef35-f19a-4cc6-851a-26376e072e76
- Downloads
-
krull_dimension_in_modal_logic(Final published version)
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