Standard fuzzy uniform structures based on continuous t-norms
Given a set \(X\), \textit{J.~Gutiérrez~García, S.~Romaguera} and \textit{M.~Sanchis} [Fuzzy Sets Syst. 161, No.~7, 1011--1021 (2010; Zbl 1206.54008)] introduced the notion of \textit{fuzzy uniform structure} on \(X\) as a collection \((\mathcal{M},\ast)\) of fuzzy pseudometrics (in the sense of~[\textit{I.~Kramosil} and \textit{J.~Michalek}, Kybernetika, Praha 11, 336--344 (1975; Zbl 0319.54002)], each of them depending on a fixed t-norm \(\ast\), which can vary from one uniform structure to another) on \(X\). Adding the respective concept of \textit{fuzzy uniformly continuous map}, the authors obtained the construct \textbf{FUNIF} of fuzzy uniform spaces. Fixing a t-norm \(\ast\), it appeared that its full subconstruct \(\mathbf{FUNIF}(\ast)\) is isomorphic to the construct \textbf{UNIF} of crisp uniform spaces in the sense of~[\textit{L.~Gillman} and \textit{M.~Jerison}, Rings of continuous functions. Reprint of the 1960 Van Nostrand edition. New York - Heidelberg - Berlin: Springer-Verlag (1976; Zbl 0327.46040)] with respect to a constructed concrete functor \(\Psi:\mathbf{FUNIF}(\ast)\rightarrow\text\textbf{UNIF}\) and its inverse \(\Phi_{\ast}:\mathbf{UNIF}\rightarrow\text\textbf{FUNIF}\). In particular, it was shown that the image under \(\Psi\) of a complete fuzzy uniform space is a complete uniform space. The main purpose of the manuscript under review is to establish that the functors in question take certain fuzzy completions to their well-known crisp motivating analogues and vice versa. More precisely, the authors show that \(\Psi\) and \(\Phi_{\ast}\) preserve precompactness (Theorem~3.6 on page~79), and therefore, also both Samuel~[\textit{P.~Samuel}, Trans. Am. Math. Soc. 64, 100--132 (1948; Zbl 0032.31401)] and Stone-Čech compactifications (Corollaries~3.15 and 3.18 on page~81, respectively). In the next step, they consider fuzzy uniformizable groups, and show that the above functors preserve local precompactness and Weil completeness (Theorems~4.7 and 4.8 on page~84, respectively), as well as providing a convenient definition of quotient fuzzy uniform structures (on quotient groups). Additionally, the authors consider the property of precompactness and completeness of the Hausdorff-Bourbaki fuzzy uniformity of a fuzzy uniform space (Theorems~5.4 and 5.5 on page~87, respectively). The last section of the paper is devoted to a brief sketch of the possible generalization technique of the obtained results to quasi-pseudometrics~[\textit{Y.~J.~Cho, M.~Grabiec} and \textit{V.~Radu}, On nonsymmetric topological and probabilistic structures. Hauppauge, NY: Nova Science Publishers (2006; Zbl 1219.54001)] and quasi-uniformities~[\textit{P.~Fletcher} and \textit{W.~F.~Lindgren}, Quasi-uniform spaces. New York Basel: Marcel Dekker (1982; Zbl 0501.54018)]. The paper has a number of non-essential typos, contains a reasonable amount of its necessary preliminaries (especially from the previous paper of the authors), is rather technical in nature, and, additionally, requires from the reader a certain (and essential) background on the theory of uniform spaces.
(locally) precompact fuzzy uniform space
scientific article; zbMATH DE number 6109296
scientific article