An extension of the notion of closure system is studied adapting the idea of meetsubsemilattice to a complete fuzzy lattice. Results relating closure operators and closure systems in the classical case are extended properly to this framework. This definition is proved to be equivalent to the most used definition given by Bělohlávek on the fuzzy powerset lattice. A definition of fuzzy closure system is presented and related to closure systems.
Atlantis Studies in Uncertainty Modelling, volume 3Joint Proceedings of the 19th World Congress of the International Fuzzy Systems Association (IFSA), the 12th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), and the 11th International Summer School on Aggregation Operators (AGOP)
Preconcepts are basic units of knowledge that form the basis of formal concepts in formal concept analysis (FCA). This paper investigates the relations among different kinds of preconcepts, such as protoconcepts, meet and join-semiconcepts and formal concepts. The first contribution of this paper, is to present a fuzzy powerset lattice gradation, that coincides with the preconcept lattice at its 1-cut. The second and more significant contribution, is to introduce a preconcept algebra gradation that yields different algebras for protoconcepts, semiconcepts, and concepts at different cuts. This result reveals new insights into the structure and properties of the different categories of preconcepts.
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