A t-norm embedding theorem for fuzzy sets
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. (C) 2012 Elsevier B.V. All rights reserved.
t-norms embedding theorem algebraic operations mathematical morphology extension principle fuzzy convex sets