A natural correspondence between quasiconcave functions and fuzzy norms

In this note we show that the usual notion of fuzzy norm defined on a linear space is equivalent to that of quasiconcave function, in the sense that every fuzzy norm N:X×R→[0,1] defined on a (real or complex) linear space X is uniquely determined by a quasiconcave function f:X→[0,1]. We explore the...

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Bibliographic Details
Published in:Fuzzy sets and systems 2023-08, Vol.466, p.108413, Article 108413
Main Authors: Cabello Sánchez, Javier, Morales González, Daniel
Format: Article
Language:English
Subjects:
Decomposition Theorem
Fuzzy normed spaces
Quasiconcave functions
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Summary:In this note we show that the usual notion of fuzzy norm defined on a linear space is equivalent to that of quasiconcave function, in the sense that every fuzzy norm N:X×R→[0,1] defined on a (real or complex) linear space X is uniquely determined by a quasiconcave function f:X→[0,1]. We explore the minimum requirements that we need to impose to some quasiconcave function f:X→[0,1] in order to define a fuzzy norm N:X×R→[0,1]. Later we use this equivalence to prove some properties of fuzzy norms, like a generalisation of the celebrated Decomposition Theorem.
ISSN:0165-0114
1872-6801
DOI:10.1016/j.fss.2022.10.005