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Open AccessArticle

Transitivity in Fuzzy Hyperspaces

1
Academia de Matemáticas, Universidad Autónoma de la Ciudad de México, Calz. Ermita Iztapalapa S/N, Col. Lomas de Zaragoza 09620, México D.F., Mexico
2
Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco 186, Col. Vicentina, Del. Iztapalapa, Mexico City C.P. 09340, Mexico
3
Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Av. Vicent Sos Baynat s/n, C.P. 12071 Castelló de la Plana, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(11), 1862; https://doi.org/10.3390/math8111862
Received: 18 September 2020 / Revised: 5 October 2020 / Accepted: 9 October 2020 / Published: 24 October 2020
Given a metric space (X,d), we deal with a classical problem in the theory of hyperspaces: how some important dynamical properties (namely, weakly mixing, transitivity and point-transitivity) between a discrete dynamical system f:(X,d)(X,d) and its natural extension to the hyperspace are related. In this context, we consider the Zadeh’s extension f^ of f to F(X), the family of all normal fuzzy sets on X, i.e., the hyperspace F(X) of all upper semicontinuous fuzzy sets on X with compact supports and non-empty levels and we endow F(X) with different metrics: the supremum metric d, the Skorokhod metric d0, the sendograph metric dS and the endograph metric dE. Among other things, the following results are presented: (1) If (X,d) is a metric space, then the following conditions are equivalent: (a) (X,f) is weakly mixing, (b) ((F(X),d),f^) is transitive, (c) ((F(X),d0),f^) is transitive and (d) ((F(X),dS)),f^) is transitive, (2) if f:(X,d)(X,d) is a continuous function, then the following hold: (a) if ((F(X),dS),f^) is transitive, then ((F(X),dE),f^) is transitive, (b) if ((F(X),dS),f^) is transitive, then (X,f) is transitive; and (3) if (X,d) be a complete metric space, then the following conditions are equivalent: (a) (X×X,f×f) is point-transitive and (b) ((F(X),d0) is point-transitive. View Full-Text
Keywords: fuzzy set; skorokhod metric; endograph metric; sendograph metric; Zadeh’s extension; Transitivity; weakly mixing; point transitivity fuzzy set; skorokhod metric; endograph metric; sendograph metric; Zadeh’s extension; Transitivity; weakly mixing; point transitivity
MDPI and ACS Style

Jardón, D.; Sánchez, I.; Sanchis, M. Transitivity in Fuzzy Hyperspaces. Mathematics 2020, 8, 1862.

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