Fuzzy convergence structures in the framework of L-convex spaces

In this paper,  fuzzy convergence theory in the framework of $L$-convex spaces is introduced. Firstly, the concept of $L$-convex remote-neighborhood spaces is introduced and it is shown that the  resulting category is isomorphic to that of $L$-convex spaces. Secondly, by means of $L$-convex ideals, the notion of $L$-convergence spaces is introduced and it is proved that the  category of $L$-convex spaces can be embedded in that of $L$-convergence spaces as a reflective subcategory.  Finally, the concepts of  convex and preconvex $L$-convergence spaces are introduced and it is shown that  the  resulting categories are isomorphic to  the  categories of $L$-convex spaces  and  $L$-preconvex remote-neighborhood spaces, respectively.