View More View Less
  • 1 Department of Mathematics and Informatics, Sofia University, Blvd. J. Bourchier 5, 1164 Sofia, Bulgaria
Restricted access

Abstract

Generalizing de Vries’ duality theorem [9], we prove that the category HLC of locally compact Hausdorff spaces and continuous maps is dual to the category DHLC of complete local contact algebras and appropriate morphisms between them.

  • [1] Adámek, J., Herrlich, H., Strecker, G. E. 1990 Abstract and Concrete Categories Wiley Interscience New York.

  • [2] Aiello, M., Pratt-Hartmann, I., Benthem van, J. (eds.) 2007 Handbook of Spatial Logics Springer-Verlag Berlin Heidelberg.

  • [3] Balbiani, Ph. (Ed.), Special issue on spatial reasoning, J. Appl. Non-Classical Logics, 12 (2002).

  • [4] Bennett, B., Düntsch, I. 2007 Axioms, algebras and topology Aiello, M., Pratt-Hartmann, I., Benthem van, J. (eds.) Handbook of Spatial Logics Springer-Verlag Berlin Heidelberg 99160 .

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [5] Cohn, A., Hazarika, S. 2001 Qualitative spatial representation and reasoning: an overview Fundam. Inform. 46 129.

  • [6] Cohn, A., Renz, J. 2008 Qualitative spatial representation and reasoning Hermelen van, F., Lifschitz, V., Porter, B. (eds.) Handbook of Knowledge Representation Elsevier Amsterdam 551596 .

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [7] Császár, Á. 1963 Foundations of General Topology MacMillan New York.

  • [8] Laguna de, T. 1922 Point, line and surface as sets of solids The Journal of Philosophy 19 449461 .

  • [9] Vries de, H. 1962 Compact Spaces and Compactifications, an Algebraic Approach Van Gorcum and Comp. N.V. Assen.

  • [10] Dimov, G. 2009 A generalization of de Vries’ duality theorem Applied Categorical Structures 17 501516 .

  • [11] Dimov, G. 2009 Some generalizations of the Fedorchuk duality theorem – I Topology Appl. 156 728746 .

  • [12] Dimov, G., Some generalizations of the Fedorchuk duality theorem – II, http://arxiv.org/abs/arXiv:0710.0181v1, 1–20.

  • [13] Dimov, G., A de Vries-type duality theorem for locally compact spaces – II, http://arxiv.org/abs/arXiv:0903.2593v4, 1–37.

  • [14] Dimov, G., A de Vries-type duality theorem for the category of locally compact spaces and continuous maps. II, Acta Math. Hungar.

  • [15] Dimov, G., Vakarelov, D. 2006 Contact algebras and region-based theory of space: a proximity approach – I Fundam. Inform. 74 209249.

    • Search Google Scholar
    • Export Citation
  • [16] Düntsch, I. (Ed.), Special issue on qualitative spatial reasoning, Fundam. Inform., 46 (2001).

  • [17] Düntsch, I., Winter, M. 2005 A representation theorem for Boolean contact algebras Theoretical Computer Science (B) 347 498512 .

  • [18] Engelking, R. 1977 General Topology PWN Warszawa.

  • [19] Efremovič, V. 1951 Infinitesimal spaces DAN SSSR 76 341343.

  • [20] Fedorchuk, V. V. 1973 Boolean δ-algebras and quasi-open mappings Sibirsk. Mat. Ž. 14 10881099 = Siberian Math. J., 14 (1973), 759–767.

    • Search Google Scholar
    • Export Citation
  • [21] Gelfand, I. M. 1939 On normed rings Doklady Akad. Nauk USSR 23 430432.

  • [22] Gelfand, I. M. 1941 Normierte Ringe Mat. Sb. 9 324.

  • [23] Gelfand, I. M., Naimark, M. A. 1943 On the embedding of normed rings into the ring of operators in Hilbert space Mat. Sb. 12 197213.

    • Search Google Scholar
    • Export Citation
  • [24] Gelfand, I. M., Shilov, G. E. 1941 Über verschiedene Methoden der Einführung der Topologie in die Menge der maximalen Ideale eines normierten Ringes Mat. Sb. 9 2539.

    • Search Google Scholar
    • Export Citation
  • [25] Grzegorczyk, A. 1960 Axiomatization of geometry without points Synthese 12 228235 .

  • [26] Johnstone, P. T. 1982 Stone Spaces Cambridge Univ. Press Cambridge.

  • [27] Leader, S. 1967 Local proximity spaces Math. Annalen 169 275281 .

  • [28] Mardeš, S. 1962 ic and P. Papic, Continuous images of ordered compacta, the Suslin property and dyadic compacta Glasnik Mat.-Fis. i Astronom. 17 325.

    • Search Google Scholar
    • Export Citation
  • [29] Mioduszewski, J., Rudolf, L. 1969 H-closed and extremally disconected Hausdorff spaces Dissert. Math. (Rozpr. Mat.) 66 152.

  • [30] Naimpally, S. A., Warrack, B. D. 1970 Proximity Spaces Cambridge University Press Cambridge.

  • [31] Pratt-Hartmann, I. 2007 First-order mereotopology Aiello, M., Pratt-Hartmann, I., Benthem van, J. (eds.) Handbook of Spatial Logics Springer-Verlag Berlin Heidelberg 1397 .

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [32] Roeper, P. 1997 Region-based topology J. Philos. Logic 26 251309 .

  • [33] Sikorski, R. 1964 Boolean Algebras Springer-Verlag Berlin.

  • [34] Simons, P. 1987 Parts. A Study in Ontology Clarendon Press Oxford.

  • [35] Randell, D. A., Cui, Z., Cohn, A. G. 1992 A spatial logic based on regions and connection Nebel, B., Swartout, W., Rich, C. (eds.) Proceedings of the 3rd International Conference Knowledge Representation and Reasoning Morgan Kaufmann Los Altos, CA 165176.

    • Search Google Scholar
    • Export Citation
  • [36] Tarski, A., Les fondements de la géométrie des corps, in: First Polish Mathematical Congress (Lwów, 1927). (English translation in: J. H. Woodger (Ed.), Logic, Semantics, Metamathematics, Clarendon Press (1956).).

    • Search Google Scholar
    • Export Citation
  • [37] Stone, M. H. 1937 Applications of the theory of Boolean rings to general topology Trans. Amer. Math. Soc. 41 375481.

  • [38] Vakarelov, D. 2007 Region-based theory of space: algebras of regions, representation theory and logics Gabbay, Dov (eds.) et al. Mathematical Problems from Applied Logics. New Logics for the XXIst Century. II. Springer-Verlag Berlin Heidelberg 267348 .

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [39] Vakarelov, D., Dimov, G., Düntsch, I., Bennett, B. 2002 A proximity approach to some region-based theories of space J. Applied Non-Classical Logics 12 527559 .

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [40] Whitehead, A. N. 1919 An Enquiry Concerning the Principles of Natural Knowledge Cambridge University Press Cambridge.

  • [41] Whitehead, A. N. 1929 Process and Reality MacMillan New York.

Acta Mathematica Hungarica
P.O. Box 127
HU–1364 Budapest
Phone: (36 1) 483 8305
Fax: (36 1) 483 8333
E-mail: acta@renyi.mta.hu

  • Impact Factor (2019): 0.588
  • Scimago Journal Rank (2019): 0.489
  • SJR Hirsch-Index (2019): 38
  • SJR Quartile Score (2019): Q2 Mathematics (miscellaneous)
  • Impact Factor (2018): 0.538
  • Scimago Journal Rank (2018): 0.488
  • SJR Hirsch-Index (2018): 36
  • SJR Quartile Score (2018): Q2 Mathematics (miscellaneous)

For subscription options, please visit the website of Springer Nature

Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation
1950
Volumes
per Year
3
Issues
per Year
6
Founder Magyar Tudományos Akadémia
Founder's
Address
H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Apr 2021 0 0 0
May 2021 0 0 0
Jun 2021 0 0 0
Jul 2021 0 0 0
Aug 2021 0 0 0
Sep 2021 0 0 0
Oct 2021 0 0 0