In this paper, the relations between M -fuzzifying cotopologies and M -fuzzifying closure systems, and the relations between M -fuzzifying convex structures and M -fuzzifying closure systems are studied, respectively. A categorical approach is provided to present these relations. It is shown that the category of M -fuzzifying cotopological spaces and the category of M -fuzzifying convex spaces are both bicoreflective subcategories of the category of M -fuzzifying closure spaces.

In this paper, the concept of strong inclusion orders between L-subsets is introduced. As a tool, it is applied to the following aspects. Firstly, the notion of algebraic L-closure operators is proposed and the resulting category is shown to be isomorphic to the category of L-convex spaces (also called algebraic L-closure spaces). Secondly, restricted L-hull operators, as generalizations of restricted hull operators, are introduced and the resulting category is also proved to be isomorphic to the category of L-convex spaces. Finally, by using the properties of strong inclusion orders, it is shown that the category of convex spaces can be embedded in the category of stratified L-convex spaces as a reflective subcategory and the concrete form of the coreflective functor from the category of L-convex spaces to the category of stratified L-convex spaces is presented.

By using the residual implication on a frame L, we develop a theory of separation axioms in the category of stratified L-generalized convergence spaces in the spirit of Lowen, i.e., we define for each space some degrees of fulfilling T0, T1, T2 and regularity axioms from a logical aspect. These degrees of separation axioms generalize the theory of separation axioms in the sense of J?ger.

In this paper, stratified L-ordered filter spaces and stratified L-ordered Cauchy spaces are introduced. It is shown that the category of stratified L-ordered filter spaces is a strongly Cartesian closed topological category and the category of stratified L-ordered Cauchy spaces is a Cartesian closed topological category. Also, the relationships among stratified L-ordered filter spaces, stratified L-ordered Cauchy spaces, stratified L-filter spaces and stratified L-Cauchy spaces are investigated.

Overlap functions, as a typical kind of binary aggregation functions, have been widely investigated from both of theoretical and applied viewpoints. In this paper, we will continue on the theoretical research on overlap functions. Firstly, we will define $O$-inclusion subsethoods to describe the inclusion degrees between fuzzy sets based on the residuum induced from an overlap function $O$. Secondly, by means of $O$-inclusion subsethoods, we will propose the concepts of $O$-convexities, algebraic $O$-closure operators and $O$-hull operators, and show that they are one-to-one corresponding. Finally, we will establish the relationship between $O$-convexities and fuzzy interval operators. These results will not only present a new perspective for theoretical research on overlap functions, but also provide a new approach to fuzzifications of convexities.

Based on a frame M , each mapping between M -fuzzifying convex spaces is equipped with some degree to be a CP mapping, a CC mapping, an isomorphic mapping and a quotient mapping. In this sense, the M -CP degree, the M -CC degree, the M -isomorphic degree and the M -quotient degree of mappings between M -fuzzifying convex spaces are obtained and investigated. It is shown that the M -CP degree and the M -CC degree can be characterized by hull operators of M -fuzzifying convex spaces. Moreover, it is proved that there are compatible relations among the M -CP degree, the M -CC degree, the M -isomorphic degree and the M -quotient degree by using some equalities and inequalities.