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 (aa 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 I-connectedness. As a useful framework for these studies, we introduce the concept of "connectivity systems". 0.
Let L be a lattice and q a convergence structure (or a topology) finer than the interval topology of L. In case of compact maximal chains and continuous lattice translations, the connected components of the space (L,q) are characterized using lattice conditions only. Moreover, lattice conditions of L are related to connectedness conditions of the order convergence space (L, o). Throughout this note, maximal chain conditions and maximal chain techniques play an important role.
Abstract. On ordered sets (posets, lattices) we regard topologies (or, more general convergence structures) which on any maximal chain of the ordered set induce its own interval topology. This construction generalizes several well-known intrinsic structures, and still contains enough to produce interesting results on for instance compactness and connectedness. The "maximal chain compatibility" between topology (convergence structure) and order is preserved by formation of arbitrary products, at least in case the involved order structures are conditionally complete lattices.
This paper is concerned with the notion of "ordered Cauchy space" which is given a simple internal characterization in Section 2. It gives a discription of the category of ordered Cauchy spaces which have ordered completions, and a construction of the "fine completion functor" on this category. Sections
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