On closure operators in fuzzy deductive systems and fuzzy algebras

The starting point of this paper is the classical well-known theorem due to G. Birkhoff, P. Hall, and J. Schmidt which establishes a one-to-one correspondence between compact closure operators, inductive closure operators, inductive closure systems, and closure operators generated in deductive systems (and generated in universal algebras, respectively). In the paper presented we make first steps in order to generalize this important theorem to the fuzzy set theory and fuzzy deductive systems (and fuzzy algebras, respectively)