The connections between three-way and classical concept lattices

The model of three-way concept lattices, a novel model for widely used three-way decisions, is an extension of classical concept lattices in formal concept analysis. This paper systematically analyses the connections between two types of three-way concept lattices (object-induced and attribute-induced three-way concept lattices) and classical concept lattices. The relationships are discussed from the viewpoints of elements, sets and orders, respectively. Furthermore, the necessary and sufficient conditions used to construct three-way concepts on the basis of classical concepts are proved, the algorithms building three-way concept lattices on the basis of classical concept lattices are presented. The obtained results are finally demonstrated and verified by examples.