Standard fuzzy uniform structures based on continuous t-norms
type: Item
instance of: scholarly article
title: Standard fuzzy uniform structures based on continuous t-norms
zbMATH Open document ID: 1255.54012
author: Javier Gutiérrez García
author: Salvador Romaguera
author: Manuel Sanchis
published in: Fuzzy Sets and Systems
publication date: 2012-11-21T00:00:00Z
full work available at URL: http://hdl.handle.net/10251/46731
review text: 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 a…
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54A40
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 18B99
zbMATH DE Number: 6109296
zbMATH Keywords: (fuzzy) (pseudo)metric
zbMATH Keywords: (fuzzy) quasi-uniform structure
zbMATH Keywords: (fuzzy) uniformizable group
zbMATH Keywords: (Hausdorff) (fuzzy) uniform structure
zbMATH Keywords: Hausdorff-Bourbaki (fuzzy) uniformity
zbMATH Keywords: (locally) precompact fuzzy uniform space
zbMATH Keywords: (quotient) topological group
zbMATH Keywords: (Samuel, Stone-Čech) compactification
zbMATH Keywords: t-norm
zbMATH Keywords: Weil completion of a topological group
reviewed by: Sergejs Solovjovs
MaRDI profile type: Publication
OpenAlex ID: W2049452732
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DOI: 10.1016/J.FSS.2011.10.008
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description: scientific article; zbMATH DE number 6109296