1. A transitive closure system comprising: a database embodied on a non-transitory data storage medium, the database comprising a cached transitive closure of a directed graph, the cached transitive closure containing transitive closure records for some or all entities in the directed graph, each transitive closure record specifying a transitive closure path from one entity to another entity, the transitive closure path containing at least one edge, each entity representing a user or group of users; at least one processor; and a transitive closure management module embodied on non-transitory computer readable medium including instructions translatable by the at least one processor to perform: receiving an identification of an entity defined in the database; interacting with a relational database management system receiving queries according to a relational query language, the relational database management system embodied on a non-transitory data storage medium, said interacting incrementally updating the cached transitive closure comprising: generating a new transitive closure for the entity from the directed graph; determining a delete transitive closure record, said determining the delete transitive closure record comprising: comparing the new transitive closure generated for the entity and the cached transitive closure in the database to determine a first transitive closure path for the entity that is not specified in the new transitive closure generated for the entity and that is specified in the cached transitive closure in the database; and selecting a cached transitive closure record identifying the first transitive closure path as the delete transitive closure record; and deleting the delete transitive closure record from the cached transitive disclosure in the database.