Spaces with fuzzy partitions and fuzzy transform
\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 distract the reader), provides most of its required preliminaries, and will be of interest to all those researchers who study categorical approaches to the theory of fuzzy sets.
scientific article; zbMATH DE number 6844131
scientific article