Consequence operators, interior operators and fuzzy relations

In this article the relationship between fuzzy consequence and fuzzy interior operators with fuzzy relations is analyzed. While the situation has been studied in depth if the fuzzy relation is a ⁎-preorder it is not the case for arbitrary fuzzy relations. If R is an arbitrary fuzzy relation, a fuzzy consequence γR and a fuzzy interior δR operators can be derived from R. Using the Representation Theorem for ⁎-preorders, from the columns and rows of R two ⁎-preorders, Rcol and Rrow, can be generated that coincide with the ⁎-preorders generated by γR and δR respectively. A duality result between them and the ⁎-coherence of γR are also shown.