Journal
KNOWLEDGE-BASED SYSTEMS
Volume 146, Issue -, Pages 181-189Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.knosys.2018.02.007
Keywords
Complete lattice; Formal concept analysis; Three-way concept; Three-way rough concept
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Funding
- National natural Science Foundation of China [11371130, 11401052]
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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. (C) 2018 Elsevier B.V. All rights reserved.
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