A t-norm embedding theorem for fuzzy sets

作者：Bielawski J^*; Tabor J

来源：Fuzzy Sets and Systems, 2012, 209: 33-53.

DOI：10.1016/j.fss.2012.06.004

• Algebraic operations
• Extension principle
• Fuzzy convex sets
• Embedding theorem
• t-Norms
• Mathematical morphology

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.

• 出版日期2012-12-16