Convex hull of a fuzzy set and triangular norms

The t-norm fuzzy sets is a class of fuzzy sets equipped with a pair of algebraic operation, a notion of T-convexity and a metric, where all these notions are based on a strict triangular norm T. In this paper we introduce in this class a notion of T-convex hull of a fuzzy set. We prove theorem that binds the T-convex hull of an upper semicontinuous fuzzy set with the convex hull of a (crisp) set. We further show two applications of this result. First, we prove that the operation of forming T-convex hull behaves well with respect to algebraic operations. Second, we show an analogue of Shapley-Folkman theorem. Shapley-Folkman theorem is a well-known result that provides an upper bound on the distance between Minkowski sum of sets and the convex hull of this sum. In this paper we show that the distance between the sum of upper semicontinuous fuzzy subsets of a finite dimensional Euclidean space and the T-convex hull of this sum has an upper bound. As a consequence of this result, we present an iterative procedure for forming T-convex hull of an upper semicontinuous fuzzy set. As an application of the results regarding t-norm fuzzy sets we show an example from a game theoretic setting, where t-norm fuzzy sets were used to handle uncertainty in a repeated two player game. (c) 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Automated summary

The paper introduces the notion of T-convex hull of fuzzy sets in t-norm fuzzy sets, proving the relationship between the T-convex hull and the convex hull as well as demonstrating two applications. It shows that the operation of forming T-convex hull behaves well with respect to algebraic operations and presents an analogue of the Shapley-Folkman theorem in the context of upper semicontinuous fuzzy subsets.