We focus on confidence-bounded association rules; we model a rather practical situation in which the confidence threshold is fixed by the user, as usually happens in applications. Within this model, we study notions of redundancy among association rules from a fundamental perspective: we discuss several existing alternative definitions and provide new characterizations and relationships between them. We show that these alternatives correspond actually to just two variants, which differ in the special treatment of full-confidence implications. For each of these two notions of redundancy, we show how to construct complete bases of absolutely minimum size.