Sublattices of Topologically Represented Lattices

In [5], a representation of bounded lattices within so-called standard topological contexts has been developed. Based on the theory of formal concept analysis [14] it includes Stone’s representation of Boolean algebras by totally disconnected compact spaces [10], Priestley’s representation of bounded distributive lattices by totally order disconnected compact spaces [7] as well as Urquhardt’s representation of bounded lattices by so-called L-spaces [11]. In the present paper we characterize the 0-1-sublattices of an arbitrary bounded lattice within its standard topological context. To do so, the concept of a closed relation of a formal context [16] is generalized to the concept of a topological relation of a topological context. This is then used to describe finite subdirect products of bounded lattices. Finally, the idea of a subdirect product construction for complete lattices [13, 16] motivates an approach to the fusion of standard topological contexts. Several examples illustrate the theoretical results.