Lattices of fuzzy objects

Abstract

The collection of fuzzy subsets of a setXforms 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, whereLis the valuation lattice– that extendsAwhile preserving all suprema and infima. The “fuzzy” objects inF(A,L)may be interpreted as the sup-preserving maps fromAto the dual ofL. In particular, each complete lattice coincides with its2-fuzzification, where2is 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 setXmay be seen as the fuzzy subsets ofXwith respect to the valuation lattice𝒫(X).