Spaces with fuzzy partitions and fuzzy transform
type: Item
instance of: scholarly article
title: Spaces with fuzzy partitions and fuzzy transform
zbMATH Open document ID: 1457.18010
author: Jiří Močkoř
published in: Soft Computing
publication date: 2018-02-27T00:00:00Z
review text: \par The paper considers a categorical approach to the setting of \(F\)-transforms of \textit{I. Perfilieva} [Fuzzy Sets Syst. 157, No. 8, 993--1023 (2006; Zbl 1092.41022)] from approximation theory. The author builds his machinery over an integral commutative quantale \(Q=(Q,\bigvee, \otimes, 1_Q)\) (see, e.g., [\textit{P. Eklund} et al., Semigroups in complete lattices. Quantales, modules and related topics. Cham: Springer (2018; Zbl 1491.06001)]). He starts with introducing the category \textbf{SpaceFP} of \textit{spaces with fuzzy partitions} (ground structures for \(F\)-transforms), i.e., pairs \((X,\mathcal{A})\), where \(X\) is a set, and \(\mathcal{A}\) is a subset of the \(Q\)-powerset \(Q^X\) such that \(\{core(A)\,|\,A\in\mathcal{A}\}\) is a partition of \(X\), with \(core(A)=\{x\in X\,|\,A(x)=1_Q\}\). The category is shown to be topological over the category \textbf{Set} of sets and maps. \par The author continues with presenting a rigorous proof of the statement that the formula of \(F\)-transform is the only choice for this type of transformation provided that this transformation satisfies some reasonable (from the approximation point of view) properties. He then introduces an isomorphic version of the category \textbf{SpaceFP}, i.e., the category \textbf{FTrans} of \textit{transformation systems}. \par The paper finishes with some functorial relationships between the category of fuzzy preordered sets and the categories of Kuratowski closure and interior operators, where one relies on a particular subcategory of the category \textbf{SpaceFP}. These functorial relationships are different from (but related to) the standard ones, which are based in fuzzy Alexandroff topology of, e.g., [\textit{P. Chen} and \textit{D. Zhang}, Fuzzy Sets Syst. 161, No. 18, 2505--2514 (2010; Zbl 1207.54014); \textit{H. Lai} and \textit{D. Zhang}, Fuzzy Sets Syst. 157, No. 14, 1865--1885 (2006; Zbl 1118.54008)]. \par The paper is not quite well written (a number of typos will…
reviewed by: Sergejs Solovjovs
Mathematics Subject Classification ID: 18F60
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 03E02
Mathematics Subject Classification ID: 06A15
Mathematics Subject Classification ID: 06A99
Mathematics Subject Classification ID: 06F07
Mathematics Subject Classification ID: 18B99
zbMATH DE Number: 6844131
zbMATH Keywords: closure operator
zbMATH Keywords: complete residuated lattice
zbMATH Keywords: core of a fuzzy set
zbMATH Keywords: \(F\)-transform
zbMATH Keywords: fuzzy partition of a set
zbMATH Keywords: fuzzy preorder relation
zbMATH Keywords: fuzzy set
zbMATH Keywords: interior operator
zbMATH Keywords: quantale
zbMATH Keywords: rough set
zbMATH Keywords: topological category
zbMATH Keywords: transformation system
MaRDI profile type: Publication
full work available at URL: s00500-017-2541-7
OpenAlex ID: W2598218473
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DOI: 10.1007/S00500-017-2541-7
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description: scientific article; zbMATH DE number 6844131