International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 4, Pages 759-766
doi:10.1155/S0161171296001056
Lattices of fuzzy objects
Department of Mathematics, Champlain Regional College, P.O. Box 5003, Lennoxville, Quebec JIM 2A1, Canada
Received 27 October 1994; Revised 8 March 1995
Copyright © 1996 Arturo A. L. Sangalli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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).