Lattice-valued convergence spaces and regularity

We define a regularity axiom for lattice-valued convergence spaces where the lattice is a complete Heyting algebra. To this end, we generalize the characterization of regularity by a ”dual form” of a diagonal condition. We show that our axiom ensures that a regular T1-space is separated and that regularity is preserved under initial constructions. Further we present an extension theorem for a continuous mapping from a subspace to a regular space. A characterization in the restricted case that the lattice is a complete Boolean algebra in terms of the closure of an L-filter is given.