House of Scientific Research (HSR). — 2014 (May 27). — 13 p. English. (OCR-слой).[Journal of Mathematics and Informatics. — Vol. 1, 2013-14, P.76-88, ISSN: 2349-0632 (P), 2349-0640 (online)].[Dong Qiu and Chong-xia Lu. College of Mathematics and Physics, Chongqing University of Posts and Telecommunication, Chongqing, China].[House of Scientific Research (HSR) is a publisher of peer-reviewed and open access journals covering a wide range of academic disciplines. The main aim of HSR is to develop highest quality knowledge-based products and service for the scientific communities all over the world].Abstract.
In this paper, we study the star-shapedness for fuzzy sets. Particularly, we clarify the exact relationships among the concepts of star-shaped fuzzy sets, quasi-starshaped fuzzy sets, pseudo-star-shaped fuzzy sets and generalized star-shaped fuzzy starshaped sets, and we obtain some important properties of these different types of starshapedness.Introduction.
The fuzzy set theory was introduced initially by Zadeh [32] in 1965. In the theory and applications of fuzzy sets, convexity plays a most useful role. From the very first, Zadeh [32] recognised its importance, and the property has been exploited in many ways involving convex fuzzy set. For example, convexity is central to the metric definitions of Klement, Puri and Ralescu [14] and Diamond and Kloeden [6, 7, 8], and to the topological properties of the corresponding metric spaces of fuzzy convex sets [10, 13].
Following the seminal work of Zadeh on the definition of a convex fuzzy set, Ammar and Metz defined another type of convex fuzzy sets in [1]. To avoid misunderstanding, Zadeh's convex fuzzy sets were called quasi-convex fuzzy sets. A lot of scholars have discussed various aspects of the theory and applications of fuzzy convex analysis.
However, Nature is not convex and, apart from possible applications, it is of independent interest to see how far the supposition of convexity can be weakened without losing too much structure. Star-shaped sets are a fairly natural extension and this note defines the notion of fuzzy star-shaped sets and explores some of their properties. In [2], Brown introduced the concept of star-shaped fuzzy sets, in [9] Diamond defined another type of star-shaped fuzzy sets (f.s., for short), and in [22] Qiu given a new type of starshaped fuzzy sets is different with the other two and established some of the basic properties of this family of fuzzy sets. In order to distinguish between these three starshaped fuzzy sets, Brown's star-shaped fuzzy sets were called quasi-star-shaped fuzzy sets (f.q-s., for short) and Qiu's star-shaped fuzzy sets were called pseudo-star-shaped fuzzy sets (f.p-s., for short). Recently, the research of fuzzy star-shaped (f.s.) sets have been again attracting the deserving attention [3, 28, 33], motivated both by Diamond's research and by the importance of the concept of fuzzy convexity [15, 16, 25, 29, 30].
In this paper, for simplicity, we consider only the star-shaped fuzzy sets defined on the Euclidean space. But it is not difficult to generalize most of the results obtained in the paper to the case that starshaped fuzzy sets are defined in linear space over real field or complex field. In Section 2, we will recall some basic concepts related to this paper and generalize star-shapedness of the normal fuzzy sets to general fuzzy sets. In Section 3, we clarify the exact relationships among the concepts of f.s. sets, f.q-s. sets and f.p-s. sets, and we will study some important properties of these different types of star-shapedness.Introduction.
Preliminaries.
Main results.
Conclusions.
Acknowledgements.
References (33 publ).
