As far as I know, there is no theory so far of functions with parts. Rather we have processes that have parts that realize (simultaneously or not) one or more functions.
Here are the cases I found in the current OBI
http://purl.obolibrary.org/obo/OBI_0000392 !'information processor function'@obi
http://purl.obolibrary.org/obo/OBI_0000393 !'signal conversion function'@obi
http://purl.obolibrary.org/obo/OBI_0000395 !'signal amplification function'@obi
http://purl.obolibrary.org/obo/OBI_0000400 !'display function'@obi; No value