49 results on '"BEZHANISHVILI, GURAM"'
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2. TEMPORAL INTERPRETATION OF MONADIC INTUITIONISTIC QUANTIFIERS.
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BEZHANISHVILI, GURAM and CARAI, LUCA
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KRIPKE semantics , *MODAL logic , *SEMANTICS (Philosophy) , *PREDICATE (Logic) - Abstract
We show that monadic intuitionistic quantifiers admit the following temporal interpretation: "always in the future" (for $\forall $) and "sometime in the past" (for $\exists $). It is well known that Prior's intuitionistic modal logic ${\sf MIPC}$ axiomatizes the monadic fragment of the intuitionistic predicate logic, and that ${\sf MIPC}$ is translated fully and faithfully into the monadic fragment ${\sf MS4}$ of the predicate ${\sf S4}$ via the Gödel translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension ${\sf TS4}$ of ${\sf S4}$ and provide a full and faithful translation of ${\sf MIPC}$ into ${\sf TS4}$. We compare this new translation of ${\sf MIPC}$ with the Gödel translation by showing that both ${\sf TS4}$ and ${\sf MS4}$ can be translated fully and faithfully into a tense extension of ${\sf MS4}$ , which we denote by ${\sf MS4.t}$. This is done by utilizing the relational semantics for these logics. As a result, we arrive at the diagram of full and faithful translations shown in Figure 1 which is commutative up to logical equivalence. We prove the finite model property (fmp) for ${\sf MS4.t}$ using algebraic semantics, and show that the fmp for the other logics involved can be derived as a consequence of the fullness and faithfulness of the translations considered. [ABSTRACT FROM AUTHOR]
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- 2023
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3. Hofmann–Mislove through the lenses of Priestley.
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Bezhanishvili, Guram and Melzer, Sebastian
- Abstract
We use Priestley duality to give a new proof of the Hofmann–Mislove Theorem. [ABSTRACT FROM AUTHOR]
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- 2022
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4. A negative solution of Kuznetsov's problem for varieties of bi-Heyting algebras.
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Bezhanishvili, Guram, Gabelaia, David, and Jibladze, Mamuka
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VARIETIES (Universal algebra) , *ALGEBRAIC varieties , *HEYTING algebras , *LOGIC , *MATHEMATICAL continuum - Abstract
In this paper, we show that there exist (continuum many) varieties of bi-Heyting algebras that are not generated by their complete members. It follows that there exist (continuum many) extensions of the Heyting–Brouwer logic H B that are topologically incomplete. This result provides further insight into the long-standing open problem of Kuznetsov by yielding a negative solution of the reformulation of the problem from extensions of I P C to extensions of H B. [ABSTRACT FROM AUTHOR]
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- 2022
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5. A New Approach to the Katětov–Tong Theorem.
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Bezhanishvili, Guram, Morandi, Patrick J., and Olberding, Bruce
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HAUSDORFF spaces , *WEIERSTRASS-Stone theorem , *COMPACT spaces (Topology) - Abstract
We give a new proof of the Katětov–Tong theorem. Our strategy is to first prove the theorem for compact Hausdorff spaces, and then extend it to all normal spaces by showing how to extend upper and lower semicontinuous real-valued functions to the Stone–Čech compactification so that the less than or equal relation between the functions is preserved. In this way, the main step of the proof is the compact case, and our approach to handling this case also leads to a new proof of a version of the Stone–Weierstrass theorem. [ABSTRACT FROM AUTHOR]
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- 2022
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6. Canonical extensions, free completely distributive lattices, and complete retracts.
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Bezhanishvili, Guram, Harding, John, and Jibladze, Mamuka
- Abstract
We provide a simple proof of a recent result of Morton and van Alten that the canonical extension of a bounded distributive lattice is its free completely distributive extension. We show that this can be used to easily obtain a number of results, both known and new. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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7. McKinsey-Tarski algebras: An alternative pointfree approach to topology.
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Bezhanishvili, Guram and Raviprakash, Ranjitha
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ALGEBRA , *TOPOLOGY , *DUALITY theory (Mathematics) , *AXIOMS , *TOPOLOGICAL spaces - Abstract
McKinsey and Tarski initiated the study of interior algebras. We propose complete interior algebras as an alternative pointfree approach to topology. We term these algebras McKinsey-Tarski algebras or simply MT-algebras. Associating with each MT-algebra the lattice of its open elements defines a functor from the category of MT-algebras to the category of frames, which we study in depth. We also study the dual adjunction between the categories of MT-algebras and topological spaces, and show that MT-algebras provide a faithful generalization of topological spaces. Our main emphasis is on developing a unified approach to separation axioms in the language of MT-algebras, which generalizes separation axioms for both topological spaces and frames. [ABSTRACT FROM AUTHOR]
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- 2023
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8. Characterization of metrizable Esakia spaces via some forbidden configurations.
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Bezhanishvili, Guram and Carai, Luca
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- 2019
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9. A generalization of de Vries duality to closed relations between compact Hausdorff spaces.
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Abbadini, Marco, Bezhanishvili, Guram, and Carai, Luca
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HAUSDORFF spaces , *COMPACT spaces (Topology) , *FUNCTION spaces , *RELATION algebras , *BOOLEAN algebra - Abstract
Stone duality generalizes to an equivalence between the categories Stone R of Stone spaces and closed relations and BA S of boolean algebras and subordination relations. Splitting equivalences in Stone R yields a category that is equivalent to the category KHaus R of compact Hausdorff spaces and closed relations. Similarly, splitting equivalences in BA S yields a category that is equivalent to the category De V S of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then gives that KHaus R is equivalent to De V S , thus resolving a problem recently raised in the literature. The equivalence between KHaus R and De V S further restricts to an equivalence between the category KHaus of compact Hausdorff spaces and continuous functions and the wide subcategory De V F of De V S whose morphisms satisfy additional conditions. This yields an alternative to de Vries duality. One advantage of this approach is that composition of morphisms is usual relation composition. [ABSTRACT FROM AUTHOR]
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- 2023
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10. On modal logics arising from scattered locally compact Hausdorff spaces.
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Bezhanishvili, Guram, Bezhanishvili, Nick, Lucero-Bryan, Joel, and van Mill, Jan
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HAUSDORFF spaces , *MODAL logic , *SEMANTICS , *SET theoretic topology , *KRULL rings - Abstract
Abstract For a topological space X , let L (X) 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 , S4. Grz n (n ≥ 1), and their intersections arise as L (X) 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 S4. Grz n for some n ≥ 1. In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space. [ABSTRACT FROM AUTHOR]
- Published
- 2019
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11. STABLE MODAL LOGICS.
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BEZHANISHVILI, GURAM, BEZHANISHVILI, NICK, and ILIN, JULIA
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MODAL logic , *MATHEMATICAL logic , *MATHEMATICAL analysis , *LOGIC , *SYLLOGISM - Abstract
Stable logics are modal logics characterized by a class of frames closed under relation preserving images. These logics admit all filtrations. Since many basic modal systems such as K4 and S4 are not stable, we introduce the more general concept of an M-stable logic, where M is an arbitrary normal modal logic that admits some filtration. Of course, M can be chosen to be K4 or S4. We give several characterizations of M-stable logics. We prove that there are continuum many S4-stable logics and continuum many K4-stable logics between K4 and S4. We axiomatize K4-stable and S4-stable logics by means of stable formulas and discuss the connection between S4-stable logics and stable superintuitionistic logics. We conclude the article with many examples (and nonexamples) of stable, K4-stable, and S4-stable logics and provide their axiomatization in terms of stable rules and formulas. [ABSTRACT FROM AUTHOR]
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- 2018
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12. MacNeille transferability and stable classes of Heyting algebras.
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Bezhanishvili, Guram, Harding, John, Ilin, Julia, and Lauridsen, Frederik Möllerström
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- 2018
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13. TYCHONOFF HED-SPACES AND ZEMANIAN EXTENSIONS OF S4.3.
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BEZHANISHVILI, GURAM, BEZHANISHVILI, NICK, LUCERO-BRYAN, JOEL, and MILL, JAN VAN
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LOGIC , *SEMANTICS , *TOPOLOGICAL spaces , *METRIC spaces , *TOPOLOGY , *HOMEOMORPHISMS - Abstract
We introduce the concept of a Zemanian logic above S4.3 and prove that an extension of S4.3 is the logic of a Tychonoff HED-space iff it is Zemanian. [ABSTRACT FROM AUTHOR]
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- 2018
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14. Canonical extensions of bounded archimedean vector lattices.
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Bezhanishvili, Guram, Morandi, Patrick J., and Olberding, Bruce
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- 2018
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15. Epimorphisms in varieties of residuated structures.
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Bezhanishvili, Guram, Moraschini, Tommaso, and Raftery, James G.
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MORPHISMS (Mathematics) , *VARIETIES (Universal algebra) , *LATTICE theory , *MONOIDS , *BROUWERIAN algebras , *HEYTING algebras - Abstract
It is proved that epimorphisms are surjective in a range of varieties of residuated structures, including all varieties of Heyting or Brouwerian algebras of finite depth, and all varieties consisting of Gödel algebras, relative Stone algebras, Sugihara monoids or positive Sugihara monoids. This establishes the infinite deductive Beth definability property for a corresponding range of substructural logics. On the other hand, it is shown that epimorphisms need not be surjective in a locally finite variety of Heyting or Brouwerian algebras of width 2. It follows that the infinite Beth property is strictly stronger than the so-called finite Beth property, confirming a conjecture of Blok and Hoogland. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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16. Proximity Biframes and Nachbin Spaces.
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Bezhanishvili, Guram and Morandi, Patrick
- Abstract
In our previous paper, in order to develop the pointfree theory of compactifications of ordered spaces, we introduced the concept of a proximity on a biframe as a generalization of the concept of a strong inclusion on a biframe. As a natural next step, we introduce the concept of a proximity morphism between proximity biframes. Like in the case of de Vries algebras and proximity frames, we show that the proximity biframes and proximity morphisms between them form a category PrBFrm in which composition is not function composition. We prove that the category KRBFrm of compact regular biframes and biframe homomorphisms is a proper full subcategory of PrBFrm that is equivalent to PrBFrm. We also show that PrBFrm is equivalent to the category PrFrm of proximity frames, and give a simple description of the concept of regularization using the language of proximity biframes. Finally, we describe the dual equivalence of PrBFrm and the category Nach of Nachbin spaces, which provides a direct way to construct compactifications of ordered spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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17. Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality.
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Bezhanishvili, Guram, Bezhanishvili, Nick, Sourabh, Sumit, and Venema, Yde
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By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras (complete Boolean algebras endowed with a proximity-like relation). We provide an alternative 'modal-like' duality by introducing the concept of a Gleason space, which is a pair ( X, R), where X is an extremally disconnected compact Hausdorff space and R is an irreducible equivalence relation on X. Our main result states that the category of Gleason spaces is equivalent to the category of compact Hausdorff spaces, and is dually equivalent to the category of de Vries algebras. [ABSTRACT FROM AUTHOR]
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- 2017
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18. Compact Hausdorff Heyting algebras.
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Bezhanishvili, Guram and Harding, John
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HAUSDORFF spaces , *HEYTING algebras , *PROFINITE groups , *TOPOLOGY , *BOOLEAN algebra - Abstract
We prove that the topology of a compact Hausdorff topological Heyting algebra is a Stone topology. It then follows from known results that a Heyting algebra is profinite iff it admits a compact Hausdorff topology that makes it a compact Hausdorff topological Heyting algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2016
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19. Forbidden configurations and subframe varieties.
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Bezhanishvili, Guram
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HEYTING algebras , *ALGEBRAIC logic , *LATTICE theory , *ALGEBRA , *CONFIGURATIONS (Geometry) - Abstract
In a series of papers, Ball, Pultr, and Sichler studied forbidden configurations in Priestley spaces and Esakia spaces. They showed, among other things, that the class of Heyting algebras whose Esakia spaces contain no copy of a given finite configuration is a variety iff the configuration is a tree. In this short note, we show that such varieties are examples of subframe varieties-the algebraic counterparts of subframe logics introduced by Fine in the 1980s. [ABSTRACT FROM AUTHOR]
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- 2016
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20. Teaching Discrete Mathematics Entirely From Primary Historical Sources.
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Barnett, Janet Heine, Bezhanishvili, Guram, Lodder, Jerry, and Pengelley, David
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STUDENT projects , *PROJECT method in teaching , *MATHEMATICS education , *HISTORICAL source material , *SEMESTER system in education - Abstract
We describe teaching an introductory discrete mathematics course entirely from student projects based on primary historical sources. We present case studies of four projects that cover the content of a one-semester course, and mention various other courses that we have taught with primary source projects. [ABSTRACT FROM AUTHOR]
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- 2016
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21. SPECTRA OF COMPACT REGULAR FRAMES.
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BEZHANISHVILI, GURAM, GABELAIA, DAVID, and JIBLADZE, MAMUKA
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HAUSDORFF spaces , *SPECTRUM analysis , *ISOMORPHISM (Mathematics) , *FUNCTION spaces , *TOPOLOGY - Abstract
By Isbell duality, each compact regular frame L is isomorphic to the frame of opens of a compact Hausdorff space X. In this note we study the spectrum Spec(L) of prime filters of a compact regular frame L. We prove that X is realized as the minimum of Spec(L) and the Gleason cover of X as the maximum of Spec(L). We also characterize zero-dimensional, extremally disconnected, and scattered compact regular frames by means of Spec(L). [ABSTRACT FROM AUTHOR]
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- 2016
22. S4.3 and hereditarily extremally disconnected spaces.
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Bezhanishvili, Guram, Bezhanishvili, Nick, Lucero-Bryan, Joel, and van Mill, Jan
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MODAL logic , *TOPOLOGY , *MATHEMATICAL research , *INTERVAL analysis , *SUBSPACES (Mathematics) - Abstract
The modal logic S4.3 defines the class of hereditarily extremally disconnected spaces (HED-spaces). We construct a countable HED-subspace X of the Gleason cover of the real closed unit interval [0,1] such that S4.3 is the logic of X. [ABSTRACT FROM AUTHOR]
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- 2015
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23. What Does "Less Than or Equal" Really Mean?
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Bezhanishvili, Guram and Pengelley, David
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CANTOR sets , *CANTOR distribution , *CARDINAL numbers , *MATHEMATICS theorems , *DISJUNCTION (Logic) - Abstract
The Cantor-Bernstein theorem is often stated as 'a ≤ b and b ≤ a imply a = b' for cardinalities. This suggestive form of the theorem may lead to a trap, into which many early 20th century mathematicians fell, unless we are very careful in interpreting ≤. The key is the subtle interplay between < and ≤. Originally, following Cantor, < was considered the primary relation, and ≤ was defined as the disjunction of < and =. However, the above suggestive form of the Cantor-Bernstein theorem requires the modern definition of ≤. The uncertainty, sometimes confusion, and evolution due to these subtleties can fascinate and motivate both us and our students today. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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24. PROXIMITY BIFRAMES AND COMPACTIFICATIONS OF COMPLETELY REGULAR ORDERED SPACES.
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BEZHANISHVILI, GURAM and MORANDI, PATRICK J.
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FRAMES (Combinatorial analysis) , *COMPACTIFICATION (Mathematics) , *TOPOLOGICAL spaces , *PARTIALLY ordered sets , *ISOMORPHISM (Mathematics) - Abstract
We generalize the concept of a strong inclusion on a biframe [Sch93] to that of a proximity on a biframe, which is related to the concept of a strong bi-inclusion on a frame introduced in [PP12b]. We also generalize the concept of a bi-compactification of a biframe [Sch93] to that of a compactification of a biframe, and prove that the poset of compactifications of a biframe L is isomorphic to the poset of proximities on L. As a corollary, we obtain Schauerte's characterization of bi-compactifications of a biframe [Sch93]. In the spatial case this yields Blatter and Seever's characterization of compactifications of completely regular ordered spaces [BS76] and a characterization of bi-compactifications of completely regular bispaces. [ABSTRACT FROM AUTHOR]
- Published
- 2015
25. De Vries powers: A generalization of Boolean powers for compact Hausdorff spaces.
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Bezhanishvili, Guram, Marra, Vincenzo, Morandi, Patrick J., and Olberding, Bruce
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TOPOLOGICAL spaces , *FUNCTION spaces , *BOOLEAN algebra , *HAUSDORFF spaces , *INDUCTION (Logic) - Abstract
We generalize the Boolean power construction to the setting of compact Hausdorff spaces. This is done by replacing Boolean algebras with de Vries algebras (complete Boolean algebras enriched with proximity) and Stone duality with de Vries duality. For a compact Hausdorff space X and a totally ordered algebra A , we introduce the concept of a finitely valued normal function f : X → A . We show that the operations of A lift to the set FN ( X , A ) of all finitely valued normal functions, and that there is a canonical proximity relation ≺ on FN ( X , A ) . This gives rise to the de Vries power construction, which when restricted to Stone spaces, yields the Boolean power construction. We prove that de Vries powers of a totally ordered integral domain A are axiomatized as proximity Baer Specker A -algebras; that is, the pairs ( S , ≺ ) , where S is a torsion-free A -algebra generated by its idempotents which is a Baer ring, and ≺ is a proximity relation on S . We introduce the category of proximity Baer Specker A -algebras and proximity morphisms between them, and prove that this category is dually equivalent to the category of compact Hausdorff spaces and continuous maps. This provides an analogue of de Vries duality for proximity Baer Specker A -algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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26. Modal Operators on Compact Regular Frames and de Vries Algebras.
- Author
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Bezhanishvili, Guram, Bezhanishvili, Nick, and Harding, John
- Abstract
In Bezhanishvili et al. () we introduced the category MKHaus of modal compact Hausdorff spaces, and showed these were concrete realizations of coalgebras for the Vietoris functor on compact Hausdorff spaces, much as modal spaces are coalgebras for the Vietoris functor on Stone spaces. Also in Bezhanishvili et al. () we introduced the categories MKRFrm and MDV of modal compact regular frames, and modal de Vries algebras as algebraic counterparts to modal compact Hausdorff spaces, much as modal algebras are algebraic counterparts to modal spaces. In Bezhanishvili et al. (), MKRFrm and MDV were shown to be dually equivalent to MKHaus, hence equivalent to one another. Here we provide a direct, choice-free proof of the equivalence of MKRFrm and MDV. We also detail connections between modal compact regular frames and the Vietoris construction for frames (Johnstone , ), discuss a Vietoris construction for de Vries algebras, and how it is linked to modal de Vries algebras. Also described is an alternative approach to the duality of MKRFrm and MKHaus obtained by using modal de Vries algebras as an intermediary. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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27. Idempotent generated algebras and Boolean powers of commutative rings.
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Bezhanishvili, Guram, Marra, Vincenzo, Morandi, Patrick, and Olberding, Bruce
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BOOLEAN functions , *ALGEBRA , *ORTHOGONAL curves , *IDEMPOTENTS , *HAUSDORFF spaces - Abstract
A Boolean power S of a commutative ring R has the structure of a commutative R-algebra, and with respect to this structure, each element of S can be written uniquely as an R-linear combination of orthogonal idempotents so that the sum of the idempotents is 1 and their coefficients are distinct. In order to formalize this decomposition property, we introduce the concept of a Specker R-algebra, and we prove that the Boolean powers of R are up to isomorphism precisely the Specker Ralgebras. We also show that these algebras are characterized in terms of a functorial construction having roots in the work of Bergman and Rota. When R is indecomposable, we prove that S is a Specker R-algebra iff S is a projective R-module, thus strengthening a theorem of Bergman, and when R is a domain, we show that S is a Specker R-algebra iff S is a torsion-free R-module. For indecomposable R, we prove that the category of Specker R-algebras is equivalent to the category of Boolean algebras, and hence is dually equivalent to the category of Stone spaces. In addition, when R is a domain, we show that the category of Baer Specker R-algebras is equivalent to the category of complete Boolean algebras, and hence is dually equivalent to the category of extremally disconnected compact Hausdorff spaces. For totally ordered R, we prove that there is a unique partial order on a Specker R-algebra S for which it is an f-algebra over R, and show that S is isomorphic to the R-algebra of piecewise constant continuous functions from a Stone space X to R equipped with the interval topology. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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28. MODAL LOGICS OF METRIC SPACES.
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BEZHANISHVILI, GURAM, GABELAIA, DAVID, and LUCERO-BRYAN, JOEL
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MODAL logic , *METRIC spaces , *FINITE model theory , *MODEL theory , *ISOMORPHISM (Mathematics) - Abstract
It is a classic result (McKinsey & Tarski, 1944; Rasiowa & Sikorski, 1963) that if we interpret modal diamond as topological closure, then the modal logic of any dense-in-itself metric space is the well-known modal system S4. In this paper, as a natural follow-up, we study the modal logic of an arbitrary metric space. Our main result establishes that modal logics arising from metric spaces form the following chain which is order-isomorphic (with respect to the ⊃ relation) to the ordinal ω + 3:$S4.Gr{z_1} \supset S4.Gr{z_2} \supset S4.Gr{z_3} \supset \cdots \,S4.Grz \supset S4.1 \supset S4.$It follows that the modal logic of an arbitrary metric space is finitely axiomatizable, has the finite model property, and hence is decidable. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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29. Proximity Frames and Regularization.
- Author
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Bezhanishvili, Guram and Harding, John
- Abstract
It is well known that the category KHaus of compact Hausdorff spaces is dually equivalent to the category KRFrm of compact regular frames. By de Vries duality, KHaus is also dually equivalent to the category DeV of de Vries algebras, and so DeV is equivalent to KRFrm, where the latter equivalence can be described constructively through Booleanization. Our purpose here is to lift this circle of equivalences and dual equivalences to the setting of stably compact spaces. The dual equivalence of KHaus and KRFrm has a well-known generalization to a dual equivalence of the categories StKSp of stably compact spaces and StKFrm of stably compact frames. Here we give a common generalization of de Vries algebras and stably compact frames we call proximity frames. For the category PrFrm of proximity frames we introduce the notion of regularization that extends that of Booleanization. This yields the category RPrFrm of regular proximity frames. We show there are equivalences and dual equivalences among PrFrm, its subcategories StKFrm and RPrFrm, and StKSp. Restricting to the compact Hausdorff setting, the equivalences and dual equivalences among StKFrm, RPrFrm, and StKSp yield the known ones among KRFrm, DeV, and KHaus. The restriction of PrFrm to this setting provides a new category StrInc whose objects are frames with strong inclusions and whose morphisms and composition are generalizations of those in DeV. Both KRFrm and DeV are subcategories of StrInc that are equivalent to StrInc. For a compact Hausdorff space X, the category StrInc not only contains both the frame of open sets of X and the de Vries algebra of regular open sets of X, these two objects are isomorphic in StrInc, with the second being the regularization of the first. The restrictions of these categories are considered also in the setting of spectral spaces, Stone spaces, and extremally disconnected spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2014
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30. Lattice subordinations and Priestley duality.
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Bezhanishvili, Guram
- Subjects
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LATTICE theory , *ORDINATION (Statistics) , *DUALITY theory (Mathematics) , *HEYTING algebras , *BOOLEAN algebra , *MATHEMATICAL proofs - Abstract
There is a well-known correspondence between Heyting algebras and S4-algebras. Our aim is to extend this correspondence to distributive lattices by defining analogues of S4-algebras for them. For this purpose, we introduce binary relations on Boolean algebras that resemble de Vries proximities. We term such binary relations lattice subordinations. We show that the correspondence between Heyting algebras and S4-algebras extends naturally to distributive lattices and Boolean algebras with a lattice subordination. We also introduce Heyting lattice subordinations and prove that the category of Boolean algebras with a Heyting lattice subordination is isomorphic to the category of S4-algebras, thus obtaining the correspondence between Heyting algebras and S4-algebras as a particular case of our approach. In addition, we provide a uniform approach to dualities for these classes of algebras. Namely, we generalize Priestley spaces to quasi-ordered Priestley spaces and show that lattice subordinations on a Boolean algebra B correspond to Priestley quasiorders on the Stone space of B. This results in a duality between the category of Boolean algebras with a lattice subordination and the category of quasi-ordered Priestley spaces that restricts to Priestley duality for distributive lattices. We also prove that Heyting lattice subordinations on B correspond to Esakia quasi-orders on the Stone space of B. This yields Esakia duality for S4-algebras, which restricts to Esakia duality for Heyting algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2013
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31. Funayama's theorem revisited.
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Bezhanishvili, Guram, Gabelaia, David, and Jibladze, Mamuka
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DISTRIBUTIVE law (Mathematics) , *LATTICE theory , *DISTRIBUTIVE lattices , *BOOLEAN algebra , *HEYTING algebras , *HOMOMORPHISMS - Abstract
Funayama's theorem states that there is an embedding e of a lattice L into a complete Boolean algebra B such that e preserves all existing joins and meets in L iff L satisfies the join infinite distributive law (JID) and the meet infinite distributive law (MID). More generally, there is a lattice embedding e: L → B preserving all existing joins in L iff L satisfies (JID), and there is a lattice embedding e: L → B preserving all existing meets in L iff L satisfies (MID). Funayama's original proof is quite involved. There are two more accessible proofs in case L is complete. One was given by Grätzer by means of free Boolean extensions and MacNeille completions, and the other by Johnstone by means of nuclei and Booleanization. We show that Grätzer's proof has an obvious generalization to the non-complete case, and that in the complete case the complete Boolean algebras produced by Grätzer and Johnstone are isomorphic. We prove that in the non-complete case, the class of lattices satisfying (JID) properly contains the class of Heyting algebras, and we characterize lattices satisfying (JID) and (MID) by means of their Priestley duals. Utilizing duality theory, we give alternative proofs of Funayama's theorem and of the isomorphism between the complete Boolean algebras produced by Grätzer and Johnstone. We also show that unlike Grätzer's proof, there is no obvious way to generalize Johnstone's proof to the non-complete case. [ABSTRACT FROM AUTHOR]
- Published
- 2013
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32. BOUNDED ARCHIMEDEAN ℓ-ALGEBRAS AND GELFAND-NEUMARK-STONE DUALITY.
- Author
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BEZHANISHVILI, GURAM, MORANDI, PATRICK J., and OLBERDING, BRUCE
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ALTERNATIVE algebras , *HAUSDORFF spaces , *BANACH spaces , *DUALITY theory (Mathematics) , *ARCHIMEDEAN property - Abstract
By Gelfand-Neumark duality, the category C"Alg of commutative C"- algebras is dually equivalent to the category of compact Hausdorff spaces, which by Stone duality, is also dually equivalent to the category ubal of uniformly complete bounded Archimedean ℓ -algebras. Consequently, C"Alg is equivalent to ubal, and this equivalence can be described through complexification. In this article we study ubal within the larger category bal of bounded Archimedean l-algebras. We show that ubal is the smallest nontrivial reective subcategory of bal, and that ubal consists of exactly those objects in bal that are epicomplete, a fact that includes a categorical formulation of the Stone-Weierstrass theorem for bal. It follows that ubal is the unique nontrivial reective epicomplete subcategory of bal. We also show that each nontrivial reective subcategory of bal is both monoreective and epireective, and exhibit two other interesting reective subcategories of bal involving Gelfand rings and square closed rings. Dually, we show that Specker R-algebras are precisely the co-epicomplete objects in bal. We prove that the category spec of Specker R-algebras is a mono-coreective subcategory of bal that is co-epireective in a mono-coreective subcategory of bal consisting of what we term ℓ -clean rings, a version of clean rings adapted to the order- theoretic setting of bal. We conclude the article by discussing the import of our results in the setting of complex "-algebras through complexification. [ABSTRACT FROM AUTHOR]
- Published
- 2013
33. Esakia Style Duality for Implicative Semilattices.
- Author
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Bezhanishvili, Guram and Jansana, Ramon
- Subjects
- *
SEMILATTICES , *DUALITY theory (Mathematics) , *HEYTING algebras , *HOMOMORPHISMS , *PARTIALLY ordered sets - Abstract
We develop a new duality for implicative semilattices, generalizing Esakia duality for Heyting algebras. Our duality is a restricted version of generalized Priestley duality for distributive semilattices, and provides an improvement of Vrancken-Mawet and Celani dualities. We also show that Heyting algebra homomorphisms can be characterized by means of special partial functions between Esakia spaces. On the one hand, this yields a new duality for Heyting algebras, which is an alternative to Esakia duality. On the other hand, it provides a natural generalization of Köhler's partial functions between finite posets to the infinite case. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
34. DEDEKIND COMPLETIONS OF BOUNDED ARCHIMEDEAN ℓ-ALGEBRAS.
- Author
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BEZHANISHVILI, GURAM, MORANDI, PATRICK J., and OLBERDING, BRUCE
- Subjects
- *
DEDEKIND rings , *MATHEMATICAL bounds , *ARCHIMEDEAN property , *COMMUTATIVE algebra , *GELFAND-Naimark theorem , *HAUSDORFF measures , *COMPACT spaces (Topology) , *C*-algebras - Abstract
All algebras considered in this paper are commutative with 1. Let baℓ be the category of bounded Archimedean ℓ-algebras. We investigate Dedekind completions and Dedekind complete algebras in baℓ. We give several characterizations for A ∈ baℓ to be Dedekind complete. Also, given A, B ∈ baℓ, we give several characterizations for B to be the Dedekind completion of A. We prove that unlike general Gelfand-Neumark-Stone duality, the duality for Dedekind complete algebras does not require any form of the Stone-Weierstrass Theorem. We show that taking the Dedekind completion is not functorial, but that it is functorial if we restrict our attention to those A ∈ baℓ that are Baer rings. As a consequence of our results, we give a new characterization of when A ∈ baℓ is a C*-algebra. We also show that A is a C*-algebra if and only if A is the inverse limit of an inverse family of clean C*-algebras. We conclude the paper by discussing how to derive Gleason's theorem about projective compact Hausdorff spaces and projective covers of compact Hausdorff spaces from our results. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
35. CANONICAL FORMULAS FOR wK4.
- Author
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BEZHANISHVILI, GURAM and BEZHANISHVILI, NICK
- Subjects
- *
LOGIC , *CONTACT transformations , *MATHEMATICAL logic , *MODEL theory , *MATHEMATICS - Abstract
We generalize the theory of canonical formulas for K4, the logic of transitive frames, to wK4, the logic of weakly transitive frames. Our main result establishes that each logic over wK4 is axiomatizable by canonical formulas, thus generalizing Zakharyaschev’s theorem for logics over K4. The key new ingredients include the concepts of transitive and strongly cofinal subframes of weakly transitive spaces. This yields, along with the standard notions of subframe and cofinal subframe logics, the new notions of transitive subframe and strongly cofinal subframe logics over wK4. We obtain axiomatizations of all four kinds of subframe logics over wK4. We conclude by giving a number of examples of different kinds of subframe logics over wK4. [ABSTRACT FROM PUBLISHER]
- Published
- 2012
- Full Text
- View/download PDF
36. De Vries Algebras and Compact Regular Frames.
- Author
-
Bezhanishvili, Guram
- Subjects
- *
ALGEBRA , *MATHEMATICS theorems , *AXIOM of choice , *EQUIVALENCE classes (Set theory) , *MATHEMATICAL analysis , *MATHEMATICS , *AXIOMATIC set theory - Abstract
By the de Vries theorem, the category DeV of de Vries algebras is dually equivalent to the category KHaus of compact Hausdorff spaces. By the Isbell theorem, the category KRFrm of compact regular frames is dually equivalent to KHaus. The proofs of both theorems employ the axiom of choice. It is a consequence of the de Vries and Isbell theorems that DeV is equivalent to KRFrm. We give a direct proof of this result, which is choice-free. In the absence of the axiom of countable dependent choice (CDC), the category KCRFrm of compact completely regular frames is a proper subcategory of KRFrm. We introduce the category cDeV of completely regular de Vries algebras, which in the absence of (CDC) is a proper subcategory of DeV, and show that cDeV is equivalent to KCRFrm. Finally, we show how the restriction of the equivalence of DeV and KRFrm works in the zero-dimensional and extremally disconnected cases. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
37. Subspaces of $${\mathbb{Q}}$$ whose d-logics do not have the FMP.
- Author
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Bezhanishvili, Guram and Lucero-Bryan, Joel
- Subjects
- *
SUBSPACES (Mathematics) , *MATHEMATICAL logic , *RATIONAL numbers , *NUMBER theory , *FINITE model theory , *TOPOLOGY , *MATHEMATICAL models - Abstract
We show that subspaces of the space $${\mathbb{Q}}$$ of rational numbers give rise to uncountably many d-logics over K4 without the finite model property. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
38. Connected modal logics.
- Author
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Bezhanishvili, Guram and Gabelaia, David
- Subjects
- *
TOPOLOGY , *MODAL logic , *BOREL sets , *SET theory , *BOOLEAN algebra , *ALGEBRAIC logic , *AXIOMS - Abstract
We introduce the concept of a connected logic (over S4) and show that each connected logic with the finite model property is the logic of a subalgebra of the closure algebra of all subsets of the real line R, thus generalizing the McKinsey-Tarski theorem. As a consequence, we obtain that each intermediate logic with the finite model property is the logic of a subalgebra of the Heyting algebra of all open subsets of R. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
39. Stone duality and Gleason covers through de Vries duality
- Author
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Bezhanishvili, Guram
- Subjects
- *
DUALITY theory (Mathematics) , *GLEASON'S theorem (Quantum theory) , *PROXIMITY spaces , *BOOLEAN algebra , *DIMENSIONAL analysis , *MATHEMATICAL analysis , *HAUSDORFF measures , *COMPACTIFICATION (Mathematics) - Abstract
Abstract: We introduce zero-dimensional de Vries algebras and show that the category of zero-dimensional de Vries algebras is dually equivalent to the category of Stone spaces. This shows that Stone duality can be obtained as a particular case of de Vries duality. We also introduce extremally disconnected de Vries algebras and show that the category of extremally disconnected de Vries algebras is dually equivalent to the category of extremally disconnected compact Hausdorff spaces. As a result, we give a simple construction of the Gleason cover of a compact Hausdorff space by means of de Vries duality. We also discuss the insight that Stone duality provides in better understanding of de Vries duality. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
40. Scattered and hereditarily irresolvable spaces in modal logic.
- Author
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Bezhanishvili, Guram and Morandi, Patrick J.
- Subjects
- *
TOPOLOGICAL spaces , *SCATTERING (Mathematics) , *BOUNDARY value problems , *BOREL sets , *TOPOLOGY - Abstract
When we interpret modal ◊ as the limit point operator of a topological space, the Gödel-Löb modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into α-slices S α, where α ranges over all ordinals. This provides topological completeness and definability results for extensions of GL. In particular, we axiomatize the modal logic of each ordinal α, thus obtaining a simple proof of the Abashidze–Blass theorem. On the other hand, when we interpret ◊ as closure in a topological space, the Grzegorczyk modal system Grz defines the class HI of hereditarily irresolvable spaces. We also give a partition of HI into α-slices H α, where α ranges over all ordinals. For a subset A of a hereditarily irresolvable space X and an ordinal α, we introduce the α-representation of A, give an axiomatization of the α-representation of A, and characterize H α in terms of α-representations. We prove that $${X \in {\bf H}_{1}}$$ iff X is submaximal. For a positive integer n, we generalize the notion of a submaximal space to that of an n-submaximal space, and prove that $${X \in {\bf H}_{n}}$$ iff X is n-submaximal. This provides topological completeness and definability results for extensions of Grz. We show that the two partitions are related to each other as follows. For a successor ordinal α = β + n, with β a limit ordinal and n a positive integer, we have $${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\beta+2n-1} \cup {\bf S}_{\beta+2n}}$$ , and for a limit ordinal α, we have $${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\alpha}}$$ . As a result, we obtain full and faithful translations of ordinal complete extensions of Grz into ordinal complete extensions of GL, thus generalizing the Kuznetsov–Goldblatt–Boolos theorem. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
41. THE MODAL LOGIC OF STONE SPACES: DIAMOND AS DERIVATIVE.
- Author
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Bezhanishvili, Guram, Esakia, Leo, and Gabelaia, David
- Subjects
- *
DIAMONDS , *SET theory , *ALGEBRAIC spaces , *MATHEMATICAL logic , *ANALYTIC spaces - Abstract
We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces is K4 and the modal logic of weakly scattered Stone spaces is K4G. As a corollary, we obtain that K4 is also the modal logic of compact Hausdorff spaces and K4G is the modal logic of weakly scattered compact Hausdorff spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
42. The universal modality, the center of a Heyting algebra, and the Blok–Esakia theorem
- Author
-
Bezhanishvili, Guram
- Subjects
- *
MODAL logic , *FIELD extensions (Mathematics) , *INTUITIONISTIC mathematics , *LATTICE theory , *AXIOM of choice , *ALGEBRA - Abstract
Abstract: We introduce the bimodal logic , which is the extension of Bennett’s bimodal logic by Grzegorczyk’s axiom and show that the lattice of normal extensions of the intuitionistic modal logic WS5 is isomorphic to the lattice of normal extensions of , thus generalizing the Blok–Esakia theorem. We also introduce the intuitionistic modal logic WS5.C, which is the extension of WS5 by the axiom , and the bimodal logic , which is the extension of Shehtman’s bimodal logic by Grzegorczyk’s axiom, and show that the lattice of normal extensions of WS5.C is isomorphic to the lattice of normal extensions of . [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
43. AN ALGEBRAIC APPROACH TO CANONICAL FORMULAS: INTUITIONISTIC CASE.
- Author
-
Bezhanishvili, Guram and Bezhanishvili, Nick
- Subjects
- *
MORPHISMS (Mathematics) , *HOMOMORPHISMS , *MATHEMATICAL functions , *SET theory , *DUALITY (Logic) , *AXIOMS , *ALGEBRA , *MATHEMATICAL formulas , *NUMERICAL analysis , *MATHEMATICAL logic - Abstract
We introduce partial Esakia morphisms, well partial Esakia morphisms, and strong partial Esakia morphisms between Esakia spaces and show that they provide the dual description of (⋀,→) homomorphisms, (⋀,→, 0) homomorphisms, and (⋀,→⋁,) homomorphisms between Heyting algebras, thus establishing a generalization of Esakia duality. This yields an algebraic characterization of Zakharyaschev's subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we obtain a new simplified proof (which is algebraic in nature) of Zakharyaschev's theorem that each intermediate logic can be axiomatized by canonical formulas. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
44. The modal logic of $${\beta(\mathbb{N})}$$.
- Author
-
Bezhanishvili, Guram and Harding, John
- Subjects
- *
TOPOLOGY , *ALGEBRA , *BOOLEAN algebra , *MATHEMATICS , *INFINITE groups - Abstract
Let $${\beta(\mathbb{N})}$$ denote the Stone–Čech compactification of the set $${\mathbb{N}}$$ of natural numbers (with the discrete topology), and let $${\mathbb{N}^\ast}$$ denote the remainder $${\beta(\mathbb{N})-\mathbb{N}}$$. We show that, interpreting modal diamond as the closure in a topological space, the modal logic of $${\mathbb{N}^\ast}$$ is S4 and that the modal logic of $${\beta(\mathbb{N})}$$ is S4.1.2. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
45. Zero-dimensional proximities and zero-dimensional compactifications
- Author
-
Bezhanishvili, Guram
- Subjects
- *
PROXIMITY spaces , *COMPACTIFICATION (Mathematics) , *TOPOLOGICAL spaces , *ISOMORPHISM (Mathematics) , *PROBLEM solving , *GEOMETRIC connections - Abstract
Abstract: We introduce zero-dimensional proximities and show that the poset of inequivalent zero-dimensional compactifications of a zero-dimensional Hausdorff space X is isomorphic to the poset of zero-dimensional proximities on X that induce the topology on X. This solves a problem posed by Leo Esakia. We also show that is isomorphic to the poset of Boolean bases of X, and derive Dwinger''s theorem that is isomorphic to as a corollary. As another corollary, we obtain that for a regular extremally disconnected space X, the Stone–Čech compactification of X is a unique up to equivalence extremally disconnected compactification of X. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
46. An algebraic approach to subframe logics. Intuitionistic case
- Author
-
Bezhanishvili, Guram and Ghilardi, Silvio
- Subjects
- *
LOGIC design , *DIGITAL electronics , *LOGIC circuits , *MATHEMATICAL analysis - Abstract
Abstract: We develop duality between nuclei on Heyting algebras and certain binary relations on Heyting spaces. We show that these binary relations are in 1–1 correspondence with subframes of Heyting spaces. We introduce the notions of nuclear and dense nuclear varieties of Heyting algebras, and prove that a variety of Heyting algebras is nuclear iff it is a subframe variety, and that it is dense nuclear iff it is a cofinal subframe variety. We give an alternative proof that every (cofinal) subframe variety of Heyting algebras is generated by its finite members. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
47. Completeness of S4 with respect to the real line: revisited
- Author
-
Bezhanishvili, Guram and Gehrke, Mai
- Subjects
- *
MATHEMATICS , *SCIENCE , *LOGIC - Abstract
We prove that S4 is complete with respect to Boolean combinations of countable unions of convex subsets of the real line, thus strengthening a 1944 result of McKinsey and Tarski (Ann. of Math. (2) 45 (1944) 141). We also prove that the same result holds for the bimodal system
S4+S5+C , which is a strengthening of a 1999 result of Shehtman (J. Appl. Non-Classical Logics 9 (1999) 369). [Copyright &y& Elsevier]- Published
- 2005
- Full Text
- View/download PDF
48. Scattered, Hausdorff-reducible, and hereditarily irresolvable spaces
- Author
-
Bezhanishvili, Guram, Mines, Ray, and Morandi, Patrick J.
- Subjects
- *
TOPOLOGICAL spaces , *HAUSDORFF measures , *MATHEMATICS - Abstract
We show that a topological space is hereditarily irresolvable if and only if it is Hausdorff-reducible. We construct a compact irreducible
T1 -space and a connected Hausdorff space, each of which is strongly irresolvable. Furthermore, we show that the three notions of scattered, Hausdorff-reducible, and hereditarily irresolvable coincide for a large class of spaces, including metric, locally compact Hausdorff, and spectral spaces. [Copyright &y& Elsevier]- Published
- 2003
- Full Text
- View/download PDF
49. Localic Krull dimension.
- Author
-
Bezhanishvili, Guram, Bezhanishvili, Nick, Lucero-Bryan, Joel, and van Mill, Jan
- Subjects
- *
KRULL rings , *TOPOLOGICAL spaces , *RING theory , *MODAL logic , *MATHEMATICAL logic - Abstract
We shall define localic Krull dimension for topological spaces. In particular, a space X has the localic Krull dimension n if n is the greatest number such that X can be mapped, via a continuous and open map, onto the n-chain seen as an Alexandroff space. We shall discuss the applications of this concept in obtaining topological completeness results in modal logic. We shall also show how the localic Krull dimension is related to the Krull dimension in ring theory. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
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