Recently, there has been a lot of interest in temporal granularity, and its applications in temporal dependency theory and data mining. Generalization hierarchies used in multi-dimensional databases and OLAP serve a role similar to that of time granularity in temporal databases, but they also apply to non-temporal dimensions, like space. In this paper, we first generalize temporal functional dependencies for non-temporal dimensions, which leads to the notion of roll-up dependency (RUD).We show the applicability of RUDs in conceptual modeling and data mining. We then indicate that the notion of time granularity used in temporal databases is generally more expressive than the generalization hierarchies in multi-dimensional databases, and show how this surplus expressiveness can be introduced in non-temporal dimensions, which leads to the formalism of RUD with negation (RUD¬). A complete axiomatization for reasoning about RUD¬ is given.