Recent theoretical insights have led to the introduction of efficient algorithms for mining closed item-sets. This paper investigates potential generalizations of this paradigm to mine closed patterns in relational, graph and network databases. Several semantics and associated definitions for closed patterns in relational data have been introduced in previous work, but the differences among these and the implications of the choice of semantics was not clear. The paper investigates these implications in the context of generalizing the LCM algorithm, an algorithm for enumerating closed item-sets. LCM is attractive since its run time is linear in the number of closed patterns and since it does not need to store the patterns output in order to avoid duplicates, further reducing memory signature and run time. Our investigation shows that the choice of semantics has a dramatic effect on the properties of closed patterns and as a result, in some settings a generalization of the LCM algorithm is not possible. On the other hand, we provide a full generalization of LCM for the semantic setting that has been previously used by the Claudien system.