Characterizations of $L$-order $L$-convex spaces

In this paper, the concepts of L-enclosed L-order space, L-order L-concave space, L-internal L-order space and L-order L-convex filter are introduced. The main results are:(1) the categories of L-order L-convex spaces, L-enclosed L-order spaces, L-order L-concave spaces and L-internal L-order spaces are isomorphic; (2) the category of L-order convergence spaces based on L-order L-convex filters is topological; (3) there is a Galois correspondence between the category of L-order convergence spaces and that of L-order L-convex spaces.