There's an RDF/OWL export of the Wikidata taxonomy available at [2] as wikidata-taxonomy.nt.gz, which can be explored with Semantic Web browsers like Protege [3].
Another fundamental relation -- "part of" (P361) [4] -- expresses mereological relationships. For (oversimplified) example: "iris part of eye", "eye part of head", "head part of body", etc. Both "subclass of" and "part of" are transitive.
A separate comment of mine in this discussion [5] describes how to traverse the "subclass of" tree in the Wikidata UI and a third-party tool called Wikidata Generic Tree. The same principle applies to the "part of" tree. The latter gets less attention, but is also quite interesting.
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1. https://www.wikidata.org/wiki/Property:P279
2. http://tools.wmflabs.org/wikidata-exports/rdf/index.php?cont...
3. http://protege.stanford.edu/