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Open AccessArticle

Topology in the Alternative Set Theory and Rough Sets via Fuzzy Type Theory

Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, NSC IT4Innovations, 30. dubna 22, 701 03 Ostrava 1, Czech Republic
Mathematics 2020, 8(3), 432; https://doi.org/10.3390/math8030432 (registering DOI)
Received: 19 February 2020 / Revised: 12 March 2020 / Accepted: 13 March 2020 / Published: 16 March 2020
(This article belongs to the Special Issue Fuzziness and Mathematical Logic )
In this paper, we will visit Rough Set Theory and the Alternative Set Theory (AST) and elaborate a few selected concepts of them using the means of higher-order fuzzy logic (this is usually called Fuzzy Type Theory). We will show that the basic notions of rough set theory have already been included in AST. Using fuzzy type theory, we generalize basic concepts of rough set theory and the topological concepts of AST to become the concepts of the fuzzy set theory. We will give mostly syntactic proofs of the main properties and relations among all the considered concepts, thus showing that they are universally valid. View Full-Text
Keywords: higher-order fuzzy logic; fuzzy type theory; alternative set theory; rough sets; indiscernibility relation; fuzzy equality higher-order fuzzy logic; fuzzy type theory; alternative set theory; rough sets; indiscernibility relation; fuzzy equality
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Novák, V. Topology in the Alternative Set Theory and Rough Sets via Fuzzy Type Theory. Mathematics 2020, 8, 432.

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