A t-norm embedding theorem for fuzzy sets

Bibliographic description

punktacja MEiN [2012 A]: 40

title:	A t-norm embedding theorem for fuzzy sets
author:	Bielawski Jakub, Tabor Jacek
journal title:	Fuzzy Sets and Systems
volume:	209
date of publication :	2012
pages:	33-53
ISSN:	0165-0114
eISSN:	1872-6801
DOI:	10.1016/j.fss.2012.06.004
language:	English
journal language:	English
abstract in English:	It is well-known that the class of upper semicontinuous normal convex fuzzy sets with compact supports can be embedded isometrically as a complete convex cone in a Banach space. We prove an analogous result for a subclass of fuzzy sets that is free from the normality limitation by exchanging the standard algebraic operations on fuzzy sets with operations based on strict t-norms. This allows us to investigate a new notion of fuzzy convexity that we call T-convexity. We show that the class of upper semicontinuous fuzzy T-convex sets with nonempty compact supports can be embedded as a closed convex cone in a Banach space. This implies that fuzzy T-convex sets satisfy the cancellation law. We discuss a possible application of the embedding theorem in mathematical morphology.
keywords in English:	Algebraic operations, Extension principle, Fuzzy convex sets, Embedding theorem, t-Norms, Mathematical morphology
affiliation:	Wydział Matematyki i Informatyki : Instytut Informatyki i Matematyki Komputerowej
type:	journal article
subtype:	academic paper

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