Generalized convexities related to aggregation operators of fuzzy sets

Susana Díaz; Esteban Induráin; Vladimír Janiš; Juan Vicente Llinares; Susana Montes

Kybernetika (2017)

  • Volume: 53, Issue: 3, page 383-393
  • ISSN: 0023-5954

Abstract

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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 ] × [ 0 , 1 ] → [ 0 , 1 ] , a fuzzy subset, say X , of the real line is said to be F -convex if for any x , y , z ∈ ℝ such that x ≤ y ≤ z , it holds that μ X ( y ) ≥ F ( μ X ( x ) , μ X ( z ) ) , where μ X : ℝ → [ 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.

How to cite

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Díaz, Susana, et al. "Generalized convexities related to aggregation operators of fuzzy sets." Kybernetika 53.3 (2017): 383-393. <http://eudml.org/doc/294466>.

@article{Díaz2017,
abstract = {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.},
author = {Díaz, Susana, Induráin, Esteban, Janiš, Vladimír, Llinares, Juan Vicente, Montes, Susana},
journal = {Kybernetika},
keywords = {fuzzy sets; convexity and its generalizations; aggregation functions; fusion operators; triangular norms},
language = {eng},
number = {3},
pages = {383-393},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Generalized convexities related to aggregation operators of fuzzy sets},
url = {http://eudml.org/doc/294466},
volume = {53},
year = {2017},
}

TY - JOUR
AU - Díaz, Susana
AU - Induráin, Esteban
AU - Janiš, Vladimír
AU - Llinares, Juan Vicente
AU - Montes, Susana
TI - Generalized convexities related to aggregation operators of fuzzy sets
JO - Kybernetika
PY - 2017
PB - Institute of Information Theory and Automation AS CR
VL - 53
IS - 3
SP - 383
EP - 393
AB - 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.
LA - eng
KW - fuzzy sets; convexity and its generalizations; aggregation functions; fusion operators; triangular norms
UR - http://eudml.org/doc/294466
ER -

References

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