How do hyperoperations like tetration exist if operations are seperate relations and not repeatitions of each other.

Question

I've run into a bit of a conflict in my fundamental understanding of concepts in math. I've always known the arithmetic operation to be extensions of each other. Multiplication is repeated addition, exponentiation is repeated multiplication, tetration is repeated exponentiation, etc. The issue is, I've also run into the idea that each of these operations is a its own separate relation (generally a 3-tuple one where two of the members are operands and one is an output). I'm a bit confused on how these two concepts don't go against each other.