Characteristics of three-way concept lattices and three-way rough concept lattices

In classical formal concept (Wille's concept) analysis, a basic theorem about concept lattices is that every concept lattice is a complete lattice and conversely, every complete lattice is isomorphic to a concept lattice. Three-way concept analysis is an extended theory of formal concept analysis. Similarly, three-way concept lattices and three-way rough lattices are also complete lattices. However, unlike the classical case, not every complete lattice arises as a three-way concept lattice (or as a three-way rough concept lattice). In this paper, we focus on characterising those complete lattices which can be represented by three-way concept lattices. In order to achieve this, we first discuss some properties of special elements such as atoms and irreducible elements, and complements of three-way concept lattices. Then we give our main theorem by displaying conditions under which any complete lattice can be realised as a three-way concept lattice. Similar results are discussed and obtained for three-way rough concept lattices.