Abstract
One of the standard axioms for Boolean contact algebras says that if a region x is in contact with the join of y and z, then x is in contact with at least one of the two regions. Our intention is to examine a stronger version of this axiom according to which if x is in contact with the supremum of some family S of regions, then there is a y in S that is in contact with x. We study a modal possibility operator which is definable in complete algebras in the presence of the aforementioned axiom, and we prove that the class of complete algebras satisfying the axiom is closely related to the class of modal KTB-algebras. We also demonstrate that in the class of complete extensional contact algebras the axiom is equivalent to the statement: every region is isolated. Finally, we present an interpretation of the modal operator in the class of the so-called resolution contact algebras.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bezhanishvili, G., N. Bezhanishvili, S. Sourabh, and Y. Venema, Irreducible equivalence relations, Gleason spaces, and de Vries duality, Applied Categorical Structures 25:381–401, 2017.
Celani, S. A., Quasi-modal algebras, Mathematica Bohemica 126(4):721–736, 2001.
Celani, S. A., Simple and subdirectly irreducibles bounded distributive lattices with unary operators, International Journal of Mathematics and Mathematical Sciences 2006:1–20, 2006.
de Vries, H., Compact Spaces and Compactifications, Van Gorcum and Comp. N.V., 1962.
Dimov, G., and D. Vakarelov, Contact algebras and region-based theory of space: A proximity approach—I, Fundamenta Informaticae 74(2–3):209–249, 2006a.
Dimov, G., and D. Vakarelov, Contact algebras and region-based theory of space: A proximity approach—II, Fundamenta Informaticae 74(2–3):251–282, 2006b.
Dimov, G. D., A de Vries-type duality theorem for the category of locally compact spaces and continuous maps. I, Acta Mathematica Hungarica 129(4):314–349, 2010.
Düntsch, I., E. O. Orłowska, and H. Wang, Algebras of approximating regions, Fundamenta Informaticae 46(1–2):71–82, 2001.
Düntsch, I., and D. Vakarelov, Region-based theory of discrete spaces: a proximity approach, Annals of Mathematics and Artificial Intelligence 49:5–14, 2007.
Düntsch, I., and M. Winter, A representation theorem for Boolean contact algebras, Theoretical Computer Science 347(3):498–512, 2005.
Galton, A., The mereotopology of discrete space, in C. Freksa, and D. M. Mark, (eds), COSIT ’99: Proceedings of the International Conference on Spatial Information Theory: Cognitive and Computational Foundations of Geographic Information Science, Springer, Berlin, 1999, pp. 251–266.
Galton, A., Qualitative Spatial Change, Oxford University Press, 2000.
Gruszczyński, R., Niestandardowe teorie przestrzeni, Wydawnictwo Naukowe Uniwersytetu Mikołaja Kopernika, 2016.
Grzegorczyk, A., Axiomatizability of geometry without points, Synthese 12(2–3):228–235, 1960.
Koppelber, S., Handbook of Boolean Algebras, volume 1, chapter General Theory of Boolean Algebras, North Holland, Amsterdam, 1989.
Naimpally, S. A., and B. D. Warrack, Proximity Spaces, vol. 59 of Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, Cambridge, 1970.
Pawlak, Z., Rough sets, International Journal of Computer and Information Sciences 11:341–356, 1982.
Picado, J., and A. Pultr, Frames and locales, in Frontiers in Mathematics, Birkhäuser, 2012.
Roeper, P., Region-based topology, Journal of Philosophical Logic 26(3):251–309, 1997.
Thron, W. J., Proximity structures and grills, Mathematische Annalen 206:35–62, 1973.
Vakarelov, D., Proximity modal logic, in M. Stokhof, and Y. Venema (eds.), Proceedings of the 11th Amsterdam Colloquium, ILLC/Department of Philosophy and University of Amsterdam, 1997, pp. 301–308.
Vakarelov, D., G. Dimov, I. Düntsch, and B. Bennett, A proximity approach to some region-based theories of space, Journal of Applied Non-Classical Logics 12(3–4):527–559, 2002.
Vakarelov, D., I. Düntsch, and B. Bennett, A note on proximity spaces and connection based mereology, in Proceedings of the International Conference on Formal Ontology in Information Systems—Volume 2001, FOIS ’01, Association for Computing Machinery, New York, NY, USA. 2001, pp. 139-150.
Worboys, M., Imprecision in finite resolution spatial data, GeoInformatica 2:257–279, 1998.
Acknowledgements
This research was funded by (a) the National Science Center (Poland), grant number 2020/39/B/HS1/00216 and (b) the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 101007627. For the purpose of Open Access, the authors have applied a CC-BY public copyright license to any Author Accepted Manuscript (AAM) version arising from this submission. The authors would like to thank to anonymous referees for their criticism that helped to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Presented by Jacek Malinowski; Received May 23, 2022.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gruszczyński, R., Menchón, P. From Contact Relations to Modal Operators, and Back. Stud Logica 111, 717–748 (2023). https://doi.org/10.1007/s11225-023-10036-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-023-10036-7