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Fuzzy interior operators

We study interior operators from the point of view of fuzzy set theory. The present approach generalizes the particular cases studied previously in the literature in two aspects. First, we use complete residuated lattices as structures of truth values thus generalizing several important cases like the classical Boolean case, (left-)continuous t-norms, MV-algebras, BL-algebras, etc. Second, and more importantly, we pay attention to graded subsethood of fuzzy sets, which turns out to play an important role. In the first part, we define, illustrate by examples and study general fuzzy interior operators. The second part is devoted to fuzzy interior operators induced by fuzzy equivalence relations (similarities).