scholarly journals KRULL DIMENSION IN MODAL LOGIC

2017 ◽  
Vol 82 (4) ◽  
pp. 1356-1386 ◽  
Author(s):  
GURAM BEZHANISHVILI ◽  
NICK BEZHANISHVILI ◽  
JOEL LUCERO-BRYAN ◽  
JAN VAN MILL
Keyword(s):  
Modal Logic ◽  
Krull Dimension ◽  
Heyting Algebras ◽  
Topological Spaces ◽  
Tychonoff Space ◽  
Dimension Functions

AbstractWe develop the theory of Krull dimension forS4-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 aT1-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 formulaszemnwhich generalize the well-known Zeman formulazem. We show that the modal logicS4.Zn:=S4+ zemnis the basic modal logic ofT1-spaces of modal Krull dimension ≤n, and we construct a countable dense-in-itselfω-resolvable Tychonoff spaceZnof modal Krull dimensionnsuch thatS4.Znis complete with respect toZn. This yields a version of the McKinsey-Tarski theorem forS4.Zn. We also show that no logic in the interval [S4n+1S4.Zn) is complete with respect to any class ofT1-spaces.

QUANTIFIED MODAL LOGIC ON THE RATIONAL LINE

2014 ◽  
Vol 7 (3) ◽  
pp. 439-454 ◽  
Author(s):  
PHILIP KREMER
Keyword(s):  
Modal Logic ◽  
Real Line ◽  
Topological Spaces ◽  
Topological Semantics ◽  
Cantor Space ◽  
Quantified Modal Logic ◽  
Propositional Modal Logic ◽  
The Family ◽  
Modal Logic S4 ◽  
Rational Line

AbstractIn the topological semantics for propositional modal logic, S4 is known to be complete for the class of all topological spaces, for the rational line, for Cantor space, and for the real line. In the topological semantics for quantified modal logic, QS4 is known to be complete for the class of all topological spaces, and for the family of subspaces of the irrational line. The main result of the current paper is that QS4 is complete, indeed strongly complete, for the rational line.

Natural extension of EF-spaces and EZ-spaces to the pointfree context

2019 ◽  
Vol 69 (5) ◽  
pp. 979-988
Author(s):  
Jissy Nsonde Nsayi
Keyword(s):  
Closed Subset ◽  
Natural Extension ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Tychonoff Space ◽  
Regular Closed

Abstract Two problems concerning EF-frames and EZ-frames are investigated. In [Some new classes of topological spaces and annihilator ideals, Topology Appl. 165 (2014), 84–97], Tahirefar defines a Tychonoff space X to be an EF (resp., EZ)-space if disjoint unions of clopen sets are completely separated (resp., every regular closed subset is the closure of a union of clopen subsets). By extending these notions to locales, we give several characterizations of EF and EZ-frames, mostly in terms of certain ring-theoretic properties of 𝓡 L, the ring of real-valued continuous functions on L. We end by defining a qsz-frame which is a pointfree context of qsz-space and, give a characterization of these frames in terms of rings of real-valued continuous functions on L.

Complexity of interpolation and related problems in positive calculi

2002 ◽  
Vol 67 (1) ◽  
pp. 397-408 ◽  
Author(s):  
Larisa Maksimova
Keyword(s):  
Modal Logic ◽  
Heyting Algebras ◽  
Logical System ◽  
Complexity Bounds ◽  
Np Completeness ◽  
Interpolation Problems ◽  
Finite Set ◽  
Modal Logic S4 ◽  
Closure Algebras ◽  
Logical Calculi

AbstractWe consider the problem of recognizing important properties of logical calculi and find complexity bounds for some decidable properties. For a given logical system L, a property P of logical calculi is called decidable over L if there is an algorithm which for any finite set Ax of new axiom schemes decides whether the calculus L + Ax has the property P or not. In [11] the complexity of tabularity, pre-tabularity. and interpolation problems over the intuitionistic logic Int and over modal logic S4 was studied, also we found the complexity of amalgamation problems in varieties of Heyting algebras and closure algebras.In the present paper we deal with positive calculi. We prove NP-completeness of tabularity, DP-hardness of pretabularity and PSPACE-completeness of interpolation problem over Int+. In addition to above-mentioned properties, we consider Beth's definability properties. Also we improve some complexity bounds for properties of superintuitionistic calculi.

A first approach to abstract modal logics

1989 ◽  
Vol 54 (3) ◽  
pp. 1042-1062 ◽  
Author(s):  
Josep M. Font ◽  
Ventura Verdú
Keyword(s):  
Modal Logic ◽  
Inference Rule ◽  
Heyting Algebras ◽  
Abstract Concepts ◽  
Algebraic Models ◽  
Abstract Logic ◽  
Sequent Systems ◽  
Abstract Logics ◽  
Set Up ◽  
Modal Systems

AbstractThe object of this paper is to make a study of four systems of modal logic (S4, S5, and their intuitionistic analogues IM4 and IM5) with the techniques of the theory of abstract logics set up by Suszko, Bloom, Brown, Verdú and others. The abstract concepts corresponding to such systems are defined as generalizations of the logics naturally associated to their algebraic models (topological Boolean or Heyting algebras, general or semisimple). By considering new suitably defined connectives and by distinguishing between having the rule of necessitation only for theorems or as a full inference rule (which amounts to dealing with all filters or with open filters of the algebras) we are able to reduce the study of a modal (abstract) logic L to that of two nonmodal logics L− and L+ associated with L. We find that L is “of IM4 type” if and only if L− and L+ are both intuitionistic and have the same theorems, and logics of type S4, IM5 or S5 are obtained from those of type IM4 simply by making classical L−, L+ or both. We compare this situation with that found in recent approaches to intuitionistic modal logic using birelational models or using higher-level sequent-systems. The treatment of modal systems with abstract logics is rather new, and in our way to it we find several general constructions and results which can also be applied to other modal systems weaker than those we study in detail.

scholarly journals C-Tychonoff and L-Tychonoff Topological Spaces

2018 ◽  
Vol 11 (3) ◽  
pp. 882-892 ◽  
Author(s):  
Samirah ALZahrani
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Tychonoff Space ◽  
Local Compactness ◽  
Compact Subspace ◽  
One To One

A topological space X is called C-Tychonoff if there exist a one-to-one function f from X onto a Tychonoff space Y such that f restriction K from K onto f(K) is a homeomorphism for each compact subspace K of X. We discuss this property and illustrate the relationships between C-Tychonoffness and some other properties like submetrizability, local compactness, L-Tychononess, C-normality, C-regularity, epinormality, sigma-compactness, pseudocompactness and zero-dimensional.

A new proof of Sahlqvist's theorem on modal definability and completeness

1989 ◽  
Vol 54 (3) ◽  
pp. 992-999 ◽  
Author(s):  
G. Sambin ◽  
V. Vaccaro
Keyword(s):  
Modal Logic ◽  
Personal Computer ◽  
Topological Spaces ◽  
Straightforward Manner ◽  
Original Proof ◽  
Complete Proof ◽  
Modal Formula ◽  
First Order ◽  
Modal Definability ◽  
General Frames

There are not many global results on modal logics. One of these is the following theorem by Sahlqvist on completeness and correspondence for a wide class of modal formulae (including many well known logics, e.g. D, T, B, S4, K4, S5, …) (see [S]).Sahlqvist's Theorem. Let A be any modal formula which is equivalent to a conjunction of formulae of the form □m(A1 → A2), where m ≥ 0, A2 is positive and A1 is obtained from propositional variables and their negations applying ∧, ∨, ♢, and □ in such a way that no positive occurrence of a variable is in a subformula of the form B1 ∨ B2 or ♢ B1 within the scope of some □. Then A corresponds effectively to a first order formula, and L + A is canonical whenever Lis a canonical logic.A formula A satisfying the above conditions is henceforth called a Sahlqvist formula. Unfortunately, till now, the only complete proof was the original proof of Sahlqvist (a proof of the correspondence half has also been given by van Benthem [vB]). It is so complicated and long that even in an advanced textbook of modal logic [HC] it has not found a place. Here, by considering general frames as topological spaces, an attitude which we developed in [TD], we give a proof of Sahlqvist's theorem simplified to such an extent that one can easily grasp the key idea on which it is based and apply the resulting algorithm to specific modal formulae in a straightforward manner, suitable even for implementation on a personal computer. This key idea also improves on previous preliminary work in the same direction (see [S1], [S2]).

scholarly journals Modal Logic Axioms Valid in Quotient Spaces of Finite CW-Complexes and Other Families of Topological Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-3
Author(s):  
Maria Nogin ◽  
Bing Xu
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Topological Spaces ◽  
Quotient Spaces ◽  
Cw Complexes

In this paper we consider the topological interpretations of L□, the classical logic extended by a “box” operator □ interpreted as interior. We present extensions of S4 that are sound over some families of topological spaces, including particular point topological spaces, excluded point topological spaces, and quotient spaces of finite CW-complexes.

scholarly journals Graph theoretic representation of rings of continuous functions

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3417-3428
Author(s):  
Bedanta Bose ◽  
Angsuman Das
Keyword(s):  
Continuous Functions ◽  
Topological Spaces ◽  
Graph Structure ◽  
Zero Set ◽  
Tychonoff Space ◽  
Graph Theoretic ◽  
Maximal Ideals ◽  
Set Intersection ◽  
Algebraic Properties ◽  
Relationship Of

In this paper, we introduce a graph structure, called zero-set intersection graph ?(C(X)), on the ring of real valued continuous functions, C(X), on a Tychonoff space X. We show that the graph is connected and triangulated. We also study the inter-relationship of cliques of ?(C(X)) and ideals in C(X) which helps to characterize the structure of maximal cliques of ?(C(X)) by different kind of maximal ideals of C(X). We show that there are at least 2c many different maximal cliques which are never graph isomorphic to each other. Furthermore, we study the neighbourhood properties of a vertex and show its connection with the topology of X and algebraic properties of C(X). Finally, it is shown that two graphs are isomorphic if and only if the corresponding rings are isomorphic if and only if the corresponding topologies are homeomorphic either for first countable topological spaces or for realcompact topological spaces.

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