Kybernetika 53 no. 3, 383-393, 2017

Generalized convexities related to aggregation operators of fuzzy sets

Susana Díaz, Esteban Induráin, Vladimír Janiš, Juan Vicente Llinares and Susana MontesDOI: 10.14736/kyb-2017-3-0383


We analyze the existence of fuzzy sets of a universe that are convex with respect to certain particular classes of fusion operators that merge two fuzzy sets. In addition, we study aggregation operators that preserve various classes of generalized convexity on fuzzy sets. We focus our study on fuzzy subsets of the real line, so that given a mapping $F: [0,1] \times [0,1] \rightarrow [0,1]$, a fuzzy subset, say $X$, of the real line is said to be $F$-convex if for any $x, y, z \in \mathbb{R}$ such that $x \le y \le z$, it holds that $\mu_X(y) \ge F(\mu_X(x),\mu_X(z))$, where $\mu_X: \mathbb{R} \rightarrow [0,1]$ stands here for the membership function that defines the fuzzy set $X$. We study the existence of such sets paying attention to different classes of aggregation operators (that is, the corresponding functions $F$, as above), and preserving $F$-convexity under aggregation of fuzzy sets. Among those typical classes, triangular norms $T$ will be analyzed, giving rise to the concept of norm convexity or $T$-convexity, as a particular case of $F$-convexity. Other different kinds of generalized convexities will also be discussed as a by-product.


fuzzy sets, triangular norms, aggregation functions, convexity and its generalizations, fusion operators


03E72, 26A51


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