It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work Fuzzy sets.
[1] Linguistics is an example of a field that utilizes fuzzy set theory.
More generally, one can use any complete lattice L in a definition of a fuzzy subset A.
An important generalization principle used in fuzzification of algebraic operations is a closure property.
A similar generalization principle is used, for example, for fuzzification of the transitivity property.