Abstract
The relations between fuzzy preorders, fuzzy consequence operators, fuzzy closure systems and co-closure systems lattice structures are studied. Given a continuous t-norm ∗, we will prove that the complete lattice of all fuzzy ∗-preorders is isomorphic with a subfamily of the complete lattice of fuzzy consequence operators. Moreover, we will show that it is possible to embed both structures in the class of fuzzy closure systems by a lattice dual monomorphism between semilattices. A solution to the following open problem: “when two families of fuzzy possible valuations produce the same fuzzy implication relation” is given. Concept of ∗-interior operator of a fuzzy consequence operator is suggested and a characterization of the previous open problem through ∗-interior operator is shown.
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