Are there equivalents of fields, groups and whatnot with hyper operations?

Question

I was curious as multiplication is really just a shorthand addition, so whats so special about it? Could we generalise to all hyper operations? Does there exist algebraic structures with these operations?

Answer 1

The answer is no. Hyperoperations are defined on $\mathbb{N}$ but are not defined on arbitrary semigroups, groups or rings. In a ring, the multiplication has no reason to be defined in terms of the addition (think of the product of two matrices, for instance).