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KRULL DIMENSION IN MODAL LOGIC

2017; Cambridge University Press; Volume: 82; Issue: 4 Linguagem: Inglês

10.1017/jsl.2017.14

1943-5886

Guram Bezhanishvili, Nick Bezhanishvili, Joel Lucero‐Bryan, Jan van Mill,

Logic, programming, and type systems

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 T 1 -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 zem n which generalize the well-known Zeman formula zem . We show that the modal logic S4.Z n := S4 + zem n is the basic modal logic of T 1 -spaces of modal Krull dimension ≤ n , and we construct a countable dense-in-itself ω -resolvable Tychonoff space Z n of modal Krull dimension n such that S4.Z n is complete with respect to Z n . This yields a version of the McKinsey-Tarski theorem for S4.Z n . We also show that no logic in the interval [ S4 n+1 S4.Z n ) is complete with respect to any class of T 1 -spaces.