The main purpose of this paper is to introduce the compatibility of $M$-fuzzifying topologies and $M$-fuzzifying convexities, define an $M$-fuzzifying topological convex space, and give a method to generate an $M$-fuzzifying topological convex space. Some characterizations of $M$-fuzzifying topological convex spaces are presented. Finally, the notion of $M$-fuzzifying weak topologies is obtaine...

In this paper,  we introduce  the notion of $M$-fuzzifying interval spaces, and discuss the relationship between $M$-fuzzifying interval spaces and $M$-fuzzifying convex structures.It is proved that  the category  {bf MYCSA2}  can be embedded in  the category  {bf  MYIS}  as a reflective subcategory, where  {bf MYCSA2} and   {bf  MYIS} denote  the category of $M$-fuzzifying convex structures of...

in this paper,  we introduce  the notion of $m$-fuzzifying interval spaces, and discuss the relationship between $m$-fuzzifying interval spaces and $m$-fuzzifying convex structures.it is proved that  the category  {bf mycsa2}  can be embedded in  the category  {bf  myis}  as a reflective subcategory, where  {bf mycsa2} and   {bf  myis} denote  the category of $m$-fuzzifying convex structures of...

Based on a completely distributive lattice $M$, base axioms and subbase axioms are introduced in $M$-fuzzifying convex spaces. It is shown that a mapping $mathscr{B}$ (resp. $varphi$) with the base axioms (resp. subbase axioms) can induce a unique $M$-fuzzifying convex structure with  $mathscr{B}$ (resp. $varphi$) as its base (resp. subbase). As applications, it is proved that bases and subbase...

Inthispaper, (L,M)-fuzzy domain finiteness and (L,M)-fuzzy restricted hull spaces are introduced, and several characterizations of the category (L,M)-CS of (L,M)-fuzzy convex spaces are obtained. Then, (L,M)-fuzzy stratified (resp. weakly induced, induced) convex spaces are introduced. It is proved that both categories, the category (L,M)-SCS of (L,M)-fuzzy stratified convex spaces and the cate...

The concept of a fuzzifying topology was given in [1] under the name L-fuzzy topology. Ying studied in [9, 10, 11] the fuzzifying topologies in the case of L = [0,1]. A classical topology is a special case of a fuzzifying topology. In a fuzzifying topology τ on a set X , every subset A of X has a degree τ(A) of belonging to τ, 0 ≤ τ(A) ≤ 1. In [4], we defined the degrees of compactness, of loca...

Based on a complete Heyting algebra, we modify the definition of lattice-valued fuzzifying convergence space using fuzzy inclusion order and construct in this way a Cartesian-closed category, called the category of L−ordered fuzzifying convergence spaces, in which the category of L−fuzzifying topological spaces can be embedded. In addition, two new categories are introduced, which are called th...

A jump system is a set of integer points with an exchange property, which is a generalization of a matroid, a delta-matroid, and a base polyhedron of an integral polymatroid (or a submodular system). Recently, the concept of M-convex functions on constant-parity jump systems is introduced by Murota as a class of discrete convex functions that admit a local criterion for global minimality. M-con...