On polygonal fuzzy sets and numbers
The main purpose (and merit) of this paper is to provide a new mathematical formalization of polygonal fuzzy numbers (FNs) in dimension \(n\in \mathbb N\) which naturally includes the most famous case of trapezoidal FNs in dimension one or piecewise linear FNs in arbitrary dimension. After some preliminaries in Section \(2\), the new formalization of this notion is introduced in Definition \(5\) (Section 3), which exactly generalizes the well-known case of a one-dimensional polygonal FN, and turns out to be a relevant subclass of normal FNs on \(\mathbb R^n\). The rest of Section 3 is devoted to some properties (like completeness) of such class, heavily based upon the general Hausdorff distance between sets. In particular, the family of polygonal FNs, based upon a fixed partition \(0=\alpha_0<\alpha_1<\ldots<\alpha_m=1\) of the real unit interval, is shown to be a convex closed cone isomorphic to a particular convex closed cone of \(\mathbb R^{2(m+1)}\). The main result of this paper is given by Theorem 23, in Section \(5\), which extends to higher dimensions and general partitions of \([0,1]\) whose maximum length tends to zero. Some results already appeared when \(n=1\) and for uniform partitions, which state that an arbitrary continuous FN may be approximated by a suitable sequence of polygonal FNs using the generalized Hausdorff \(d{_\infty}\)-distance. By means of this class, the authors also show that some embedding theorems may be decisively simplified.
scientific article; zbMATH DE number 6112232
scientific article