In order to increase the expressiveness of XML query languages we define an XML object model that provides a rich description of XML data using the notions of class schema and class tuple. Based on this model we develop a novel query language, the XML Identity Query Language (XIQL) based on identifier generation and fixed-point computation. However, XIQL cannot deal with finite tree duplicates in an XML object graph. To solve this problem we extend XIQL by integrating terms denoting hereditarily finite sets. With respect to determinate transformations the extended XIQL⁺ is still incomplete, but becomes complete with respect to a modified completeness criterion for XML database transformations, which aims at capturing more non-determinism.