Abstract
To obtain the group properties of fuzzy quantities, Mares introduced an equivalence relation between fuzzy quantities. However, the Mares's method used to prove his main theorems demands to limit the investigation to fuzzy quantities with finite support. In this paper, we discuss the properties of symmetric fuzzy numbers, show an equivalent characterization of convex fuzzy sets, and present a way to construct a symmetric convex fuzzy set with a convex fuzzy set. Based on these results, we restrict ourselves to fuzzy numbers and prove Mares's results without the limitation to fuzzy numbers with finite support using the refined equivalence relation due to Hong and Do. Our results prove one of Mares's open questions in the literature.
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