A considerable amount of research has been done on the notions of pseudo complement, intersection and union of fuzzy sets [1], [4], [11]. Most of this work consists of generalizations or alternatives of the basic concepts introduced by L. A. Zadeh in his famous paper [13]: generalization of the unit interval to arbitrary complete and completely distributive lattices or to Boolean algebras [2]; alternatives to union and intersection using the concept of t-norms [3], [10]; alternative complements in the unit interval in [4], [11], a.s.o. Although it is usually accepted that I := [0,1] is a natural generalization of {0,1} and that a --> 1 - a is a natural generalization of the Boolean complement on {0,1}, we do however not find canonical and mathematical justification for this fact, which nevertheless lies at the heart of the definition of fuzzy sets. It is the purpose of this note to present a canonical way of obtaining I and L. A. Zadeh's pseudocomplement.
Moreover, if we consistently use this canonical machine we shall see that also other set-concepts can, and maybe should have been extended to fuzzy sets. Further we also give a possible generalization involving a choice of arbitrary t-norms.
@article{urn:eudml:doc:38899,
title = {On the fundamentals of fuzzy sets.},
journal = {Stochastica},
volume = {8},
year = {1984},
pages = {157-169},
zbl = {0577.04001},
mrnumber = {MR0783403},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:38899}
}
Lowen, Robert. On the fundamentals of fuzzy sets.. Stochastica, Tome 8 (1984) pp. 157-169. http://gdmltest.u-ga.fr/item/urn:eudml:doc:38899/