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Proximity Spaces of Exact Sets

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Book cover Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing (RSFDGrC 2005)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 3641))

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Abstract

[4] placed an approximation space (U,≡ ) in a type-lowering retraction with its power set 2U such that the ≡ -exact subsets of U comprise the kernel of the retraction, where ≡ is the equivalence relation of set-theoretic indiscernibility within the resulting universe of exact sets. Since a concept thus forms a set just in case it is ≡ -exact, set-theoretic comprehension in (U,≡ ) is governed by the method of upper and lower approximations of Rough Set Theory. Some central features of this universe were informally axiomatized in [3] in terms of the notion of a Proximal Frege Structure and its associated modal Boolean algebra of exact sets. The present essay generalizes the axiomatic notion of a PFS to tolerance (reflexive, symmetric) relations, where the universe of exact sets forms a modal ortho-lattice. An example of this general notion is provided by the tolerance relation of “matching” over U.

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Authors and Affiliations

  1. Department of Philosophy, The University of Pretoria, 0002, Pretoria, South Africa

    Peter John Apostoli & Akira Kanda

Authors
  1. Peter John Apostoli
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  2. Akira Kanda
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Regina, Regina, SK, S4S 0A2 Canada, Polish-Japanese Institute of Information Technology, Koszykowa 86, 02-008 Warsaw, P.O. Box, Poland

    Dominik Ślęzak

  2. School of Information Science and Technology, Southwest Jiaotong University, 610031, Chengdu, P.R. China

    Guoyin Wang

  3. Institute of Mathematics, Warsaw University, Banacha 2, 02-097, Warsaw, Poland

    Marcin Szczuka

  4. Department of Computer Science, Brock University, St. Catharines, L2S 3A1, Ontario, Canada

    Ivo Düntsch

  5. Department of Computer Science, University of Regina, S4S 0A2, Regina, Saskatchewan, Canada

    Yiyu Yao

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Apostoli, P.J., Kanda, A. (2005). Proximity Spaces of Exact Sets. In: Ślęzak, D., Wang, G., Szczuka, M., Düntsch, I., Yao, Y. (eds) Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing. RSFDGrC 2005. Lecture Notes in Computer Science(), vol 3641. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11548669_10

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  • DOI: https://doi.org/10.1007/11548669_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28653-0

  • Online ISBN: 978-3-540-31825-5

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