Elsevier

Fuzzy Sets and Systems

Volume 209, 16 December 2012, Pages 33-53
Fuzzy Sets and Systems

A t-norm embedding theorem for fuzzy sets

https://doi.org/10.1016/j.fss.2012.06.004Get rights and content

Abstract

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.

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