(L,M) -fuzzy convex structures

Abstract

In this paper, the notion of (L, M)-fuzzy convex structures is introduced. It is a generalization of L-convex structures and M-fuzzifying convex structures. In our definition of (L, M)-fuzzy convex structures, each L-fuzzy subset can be regarded as an L-convex set to some degree. The notion of convexity preserving functions is also generalized to lattice-valued case. Moreover, under the framework of (L, M)-fuzzy convex structures, the concepts of quotient structures, substructures and products are presented and their fundamental properties are discussed. Finally, we create a functor ω from MYCS to LMCS and show that MYCS can be embedded in LMCS as a coreflective subcategory, where MYCS and LMCS denote the category of M-fuzzifying convex structures and the category of (L, M)-fuzzy convex structures, respectively.

(L,M)-fuzzy topological-convex spaces

“…Definition 2.5. ( [19]) An (L, M)-fuzzy closure structure C : L X → M is called an (L, M)-fuzzy convex structure and the pair (X, C) is called an (L, M)-fuzzy convex space, if C further satisfies…”

“…Many subsequent studies have been done [23,27,28,[31][32][33]. Further, Shi and Xiu introduced (L, M)-fuzzy convex structure which is a unified form of L-convex structure and M-fuzzifying convex structure [19]. Based on this concept, many characterizations and related theories have been discussed [11,13,20,25,30].…”

On (L,M)-fuzzy convex structures

“…Also, he studied a fuzzy topology together with a fuzzy convexity on the same underlying set X, and introduced fuzzy topology fuzzy convexity spaces and the notion of fuzzy local convexity. By framework, which proposed in [23], Li [9] presented a categorical approach to enrich (L, M)-fuzzy convex structures, Xiu et al [32] presented a degree approach to study the relationship between (L, M)-fuzzy convex structures and (L, M)-fuzzy closure systems and Wu and Li [31] introduced (L, M)-fuzzy domain finiteness, (L, M)-fuzzy restricted hull spaces and several characterizations of the category (L, M)-CS of (L, M)-fuzzy convex spaces. Recently, there has been significant research on fuzzy convex structures ( [8, 13-17, 22, 28, 33, 34]).…”

Degrees of L-Continuity for Mappings between L-Topological Spaces

“…Since Chang [1] introduced fuzzy set theory to topology, fuzzy topology and its related theories have been widely investigated such as [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The degree approach that equips fuzzy topology and its related structures with some degree description is also an essential character of fuzzy set theory.…”