Connectivity in Lattice-Ordered Spaces

The classical order-theoretical characterizations of compact and connected chains, respectively, are extended to wider classes of lattices, using the fact that compactness and (path-) connectedness of maximal chains are closely related to the corresponding properties of the whole lattice (as was already pointed out in an earlier paper due to the second author). Here we replace maximal chains by “links” and study several new types of connectedness in ordered convergence spaces, such as path-connectedness, link-connectedness and 1-connectedness. As a useful framework for these studies, we introduce the concept of “connectivity systems”.