Compactness in lattice-valued function spaces

We study conditions which ensure the compactness and weak relative compactness of a set of mappings between two lattice-valued convergence spaces. We consider lattice-valued pointwise convergence and lattice-valued continuous convergence. A suitable notion of even continuity for subsets of a function space is introduced which allows to pass from compactness with respect to the lattice-valued pointwise convergence to compactness with respect to lattice-valued continuous convergence. It is shown that with a regularity condition for the second space, we can deduce compactness conditions for sets of continuous mappings between two lattice-valued convergence spaces.