Gähler's neighborhood condition for lattice-valued convergence spaces

We generalize a neighborhood condition from the category of convergence spaces to the category of lattice-valued convergence spaces. For a space in the category of convergence spaces, this condition is equivalent to being a topological space. It turns out that there are two meaningful generalizations of this condition to the category of lattice-valued convergence spaces. The first one guarantees that a space in a certain subcategory is a lattice-valued topological space. The other one is a levelwise condition which is equivalent to a generalization of Fischer's diagonal condition. The latter generalization is preserved under the embedding of the categories of convergence approach spaces and of probabilistic limit spaces into the category of lattice-valued convergence spaces.