Regular coverings in filter and ideal lattices

Abstract

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.