International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 4, Pages 759-766

Lattices of fuzzy objects

Arturo A. L. Sangalli

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.


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).