Geodesic
Generalized convexities related to aggregation operators of fuzzy sets
Kybernetika, Tome 53 (2017) no. 3, pp. 383-393
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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.
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.
DOI : 10.14736/kyb-2017-3-0383
Classification : 03E72, 26A51
Keywords: fuzzy sets; convexity and its generalizations; aggregation functions; fusion operators; triangular norms 
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     title = {Generalized convexities related to aggregation operators of fuzzy sets},
     journal = {Kybernetika},
     pages = {383--393},
     year = {2017},
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Díaz, Susana; Induráin, Esteban; Janiš, Vladimír; Llinares, Juan Vicente; Montes, Susana. Generalized convexities related to aggregation operators of fuzzy sets. Kybernetika, Tome 53 (2017) no. 3, pp. 383-393. doi: 10.14736/kyb-2017-3-0383

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