International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 4, Pages 759-766
doi:10.1155/S0161171296001056

Lattices of fuzzy objects

Abstract

The collection of fuzzy subsets of a set X forms a complete lattice that extends the complete lattice 𝒫(X) of crisp subsets of X. In this paper, we interpret this extension as a special case of the “fuzzification” of an arbitrary complete lattice A. We show how to construct a complete lattice F(A,L) –the L-fuzzificatio of A, where L is the valuation lattice– that extends A while preserving all suprema and infima. The “fuzzy” objects in F(A,L) may be interpreted as the sup-preserving maps from A to the dual of L. In particular, each complete lattice coincides with its 2-fuzzification, where 2 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 set X may be seen as the fuzzy subsets of X with respect to the valuation lattice 𝒫(X).