  The Dedekind-Birkhoff theorem for finite-height modular lattices has
previously been generalized to complete modular lattices using the theory of
regular coverings. In this paper, we investigate regular coverings in lattices
of filters and lattices of ideals, and the regularization strategy--embedding
the lattice into its lattice of filters or lattice of ideals, thereby possibly
converting a covering which is not regular into a covering which is regular.
One application of the theory is a generalization of the notion of chief
factors, and of the Jordan-Holder Theorem, to cases where the modular lattice
in question is of infinite height. Another application is a formalization of
the notion of the steps in the proof of a theorem.
