On the relation between fuzzy preorders and fuzzy consequence operators

The purpose of this paper is to analyze the operators induced by relations and conversely the relations induced by operators in fuzzy logic. Given a t-norm * and given a non-empty universal set X, it is well known that if R is a fuzzy *-preorder on X then the operator induced by R, C-R*, is a fuzzy consequence operator (FCO). In fact, C-R* is a *-coherent FCO. It is also known that if C is a *-coherent FCO then the relation induced by C, R-C, is a fuzzy *-preorder. We explore the *-coherence axiom because we do not know in the literature any example of a non-coherent operator. Then, several families of these operators will be shown. Moreover we prove that the equivalence between fuzzy preorders and fuzzy consequence operators is held in only one way. As a result, a characterization of the *-preolder concept using the induced operator is given. Also some characterizations which show when an operator induces a e-preorder are proved. Finally, we will show that the characterization of the operators induced by relations given for finite universes cannot be generalized for infinite universes.