Lattices of fuzzy objects. (1996) - Sangalli, Arturo A. L.
Abstract : The collection of fuzzy subsets of a setX forms a complete lattice that extends the complete lattice𝒫 ( X ) of crisp subsets ofX . In this paper, we interpret this extension as a special case of the fuzzification of an arbitrary complete latticeA . We show how to construct a complete latticeF ( A, L ) theL -fuzzificatio ofA, whereL is the valuation lattice that extendsA while preserving all suprema and infima. The fuzzy objects inF ( A, L ) may be interpreted as the sup-preserving maps fromA to the dual ofL . In particular, each complete lattice coincides with its2 -fuzzification, where2 is the twoelement lattice. Some familiar fuzzifications (fuzzy subgroups, fuzzy subalgebras, fuzzy topologies, etc.) are special cases of our construction. Finally, we show that the binary relations on a setX may be seen as the fuzzy subsets ofX with respect to the valuation lattice𝒫 ( X ) .