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The study of algebraic structures and properties applying to large classes of such structures. For example, ideas from group theory and ring theory are extended and considered for structures with other signatures (systems of basic or fundamental operations).

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Is there a term for operations that are only defined for some members of a set?

You might be looking for partial functions: https://en.m.wikipedia.org/wiki/Partial_function
D. Zack Garza's user avatar