On fuzzy pseudo-normed vector spaces
type: Item
instance of: scholarly article
title: On fuzzy pseudo-normed vector spaces
zbMATH Open document ID: 0682.46004
DOI: 10.1016/0165-0114(88)90061-9
author: Nehad N. Morsi
published in: Fuzzy Sets and Systems
publication date: 1988-01-01T00:00:00Z
review text: The author defines a fuzzy (pseudo-)metric on a set by assigning to each pair of points a nonnegative member of the fuzzy real line of \textit{B. Hutton} [J. Math. Anal. Appl. 50, 74-79 (1975; Zbl 0297.54003)] with the real unit interval as the range space for fuzzy sets. The operation of addition used in the triangle inequality is based on Zadeh's extension principle. The author defines fuzzy open balls which form a base for a fuzzy topology. The definition of a fuzzy (pseudo-)metric is of the same nature as the one of a fuzzy (pseudo-)norm introduced by \textit{U. Höhle} [in: ``Fuzzy sets'': Theory Appl. to Policy Anal.'', New York 1980, 13-24 (1980; Zbl 0611.54006)]. The author continues the study of Höhle's (pseudo-)normed spaces. He shows that a fuzzy (pseudo-)norm generates (in a very well-known way) a (pseudo-)metric in the author's sense. He then shows that the algebraic operations in a (pseudo-)normed space are continuous with respect to the fuzzy topology induced by the (pseudo- )norm (the fuzzy topology on the real or complex field is the one consisting of all lower semi-continuous functions), and that the fuzzy open balls at zero are convex, balanced and absorbing in the sense of \textit{A. K. Katsaras} and \textit{D. B. Liu} [J. Math. Anal. Appl. 58, 135- 146 (1977; Zbl 0358.46011)]. There is also shown that each fuzzy (pseudo- )norm generates a fuzzy semi-norm of \textit{A. K. Katsaras} [Fuzzy Sets Syst. 12, 143-154 (1984; Zbl 0555.46006)] and vice versa, and that the resulting fuzzy topologies then coincide.
Mathematics Subject Classification ID: 46A99
Mathematics Subject Classification ID: 54A40
zbMATH DE Number: 4117205
zbMATH Keywords: pseudo-norm
zbMATH Keywords: fuzzy (pseudo-)metric
zbMATH Keywords: triangle inequality
zbMATH Keywords: Zadeh's extension principle
zbMATH Keywords: fuzzy open balls
zbMATH Keywords: fuzzy topology
zbMATH Keywords: Höhle's (pseudo- )normed spaces
zbMATH Keywords: lower semi-continuous functions
reviewed by: Tomasz Kubiak
MaRDI profile type: Publication
full work available at URL: 0165-0114(88)90061-9
OpenAlex ID: W2015651254
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description: scientific article; zbMATH DE number 4117205