Set theory is the mathematical context in which relations of taxonomy and hyponymy are defined. Set theory defines a hierarchy of concepts of sets, set systems, sets of sets of sets, etc. Sets and proper classes are taxonomically related to one another if they are subsets or subclasses of one another. So taxonomy can be described by the mathematical theory of set inclusion.
Hyponomy is distinguished from taxonomy by the fact that it describes instances rather then subclasses. So the instances of hyponymy relations need not be classes themselves. Hyponomy is particularly interesting when it is applied to sets and classes. To give an example, the class of antisymmetric binary relations is a taxonomical subclass of the class of binary relations and it is an hyponym of the class of subclass closed families.
A mathematical taxonomy will include all the different sets and classes of mathematical structures, elements, and expressions that any given mathematical theory will have. A mathematical taxonomy is provided by graphclasses, but otherwise there does not seem to be a general mathematical taxonomy available. The mathematical taxonomy could be enriched with hyponomy and related relations to create a general formal ontology of mathematics.
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