Lattice-theoretic three-way formal contexts and their concepts.

Three-way concept analysis provides a novel model for three-way decisions and also extends the classical formal concept analysis, which has attracted many attentions in recent years. By forgetting the point set, many mathematical structures related to sets can be extended to certain structures of powersets and consequently be studied as ordered structures; research in this area has yielded some interesting results. In order to study three-way concept analysis in the ordered structures, in this paper, a kind of lattice-theoretic formal contexts is proposed unprecedentedly. Firstly, we introduce the lattice-theoretic three-way formal concept lattices. Secondly, the relationship between lattice-theoretic three-way formal concepts and the lattice-theoretic formal concepts is investigated. Thirdly, we illustrate that the lattice-theoretic three-way formal concept lattice is a direct generalization of L-fuzzy formal concepts based on three-way decisions. Fourthly, the fundamental theorem for the special cases of lattice-theoretic three-way formal contexts is studied. Finally, we show that the lattice-theoretic three-way concepts are pointfree forms of some kinds of classical concepts; moreover, by the fundamental theorem, we get the characterization theorems for the classical three-way concept lattices, three-way rough concepts lattices and mixed concepts lattices; these results are compared with the available ones. [ABSTRACT FROM AUTHOR]