Convergence structures in L-concave spaces

Considering a complete residuated lattice L as the lattice background, the concept of (preconcave, concave) L-convergence spaces via L-ordered co-Scott closed sets is introduced and its diagonal axioms are proposed. It is shown that concave L-convergence spaces are isomorphic to strong L-concave spaces in a categorical viewpoint. Also, it is proved that a preconcave L-convergence space satisfies the Kowalsky diagonal axiom if and only if it is concave, and an L-convergence space satisfies the Fischer diagonal axiom if and only if it is concave.