A new approach to the fuzzification of arity, JHC and CUP of L-convexities
What is it about?
In this paper, the notions of arity, Corn-union property (CUP), Join-hull commutativity (JHC) and weakly Join-hull commutativity (weakly JHC) are generalized to the fuzzy case. Also, it is proved that Corn-union property, Join-hull commutativity and weakly Join-hull commutativity L-convexities are of arity ≤2. Moreover, the CP and CC mappings between L-convexities with arity property are characterized.
Why is it important?
Convexity, being originally inspired by the shape of some figures, such as circles and polyhedrons in Euclidean spaces, has been applied to many areas in the study of extremum problems. In fact, convexity exists in so many mathematical research areas, such as lattices [6, 34], algebras [14, 17], metric spaces [16], graphs [4, 5, 9], matroids [28, 29, 38] and topological spaces [12, 35]. In 1993, M. van de Vel collected the theory of convexity systematically in the famous book [36]. As we know, matroid theory plays an important role in combinatorial optimization problems [38]. Many real-world problems can be defined and solved by making use of matroid theory. In fact, a matroid is a convex structure that satisfies the Exchange Law (see Chapter I, Section 2 in [36] for detail). In this case, extending the theory of convex structures to the fuzzy setting is a particularly meaningful topic in areas of both theoretical research and practical application. In 1994, Rosa [23] firstly generalized convex spaces to the fuzzy situation. Based on a completely distributive lattice L, Maruyama [15] defined another more generalized fuzzy convex spaces, which is called L-fuzzy convex spaces. In the sense of Rosa and Maruyama, each convex set was fuzzy, but the convex structure comprised by those fuzzy convex sets is crisp. Recently, Pang and Zhao [18] presented the notions of L-concave structures, concave L-interior operators and concave L-neighborhood systems, and it is proved that they are all isomorphic to L-convex structures. Later, Pang and Shi [19] proposed several types of L-convex spaces and established their mutual categorical relations. Subsequently, many properties of Convexity theory are generalized to the L-fuzzy case [13, 20, 21, 25, 41, 42]. In a different way, Shi and Xiu [32] provided a new approach to the fuzzification of convex spaces and presented the notion of M-fuzzifying convex structures, where each subset can be regarded as a convex set to some degree. Further, many properties of M-fuzzifying convex structures have been investigated in depth such as M-fuzzifying restricted hull operators [31], M-fuzzifying Join-Hull-Commutativity [39], M-fuzzifying interval operators [43] and geometric property of M-fuzzifying convex structures [40]. Afterwards, abstract convexity was extended to a more general case, which is called (L, M)-fuzzy convexity in [33].