L-(resp. concave) down-directed convergence relation spaces and L-(resp. concave) filter convergence spaces

Convergence structure and relation are useful tools in interpreting many mathematical structures such as topological spaces and convex spaces. The aim of this paper is to study convergence structures in the framework of L-concave spaces by using relations. Specifically, the notion of L-down-directed relations is introduced and some simple examples are presented. Based on this, notions of L-down-directed convergence relation spaces and L-concave down-directed convergence relations are introduced. It is proved that the category of L-concave internal relation spaces can be embedded into the category of L-down-directed convergence relation spaces as a reflective subcategory. In addition, the category of L-concave down-directed convergence relation spaces is isomorphic to the category of L-concave internal relation spaces. In order to characterize L-down-directed convergence relation space and L-concave down-directed convergence relation space, notions of L-concave filters, L-filter convergence spaces and L-concave filter convergence spaces are introduced. It is showed that the category of L-down-directed convergence relation spaces is isomorphic to the category of L-filter convergence spaces. It also showed that the category of L-concave down-directed convergence relation spaces is isomorphic to the category of L-concave filter convergence spaces and the category of L-concave spaces.