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).
