The set of integers, $\mathbb{Z}$ is an extension of natural numbers $\mathbb{N}=\{0, 1, 2, \cdots\}$ that allows us to solve equations of the form $x+a=b.$ Similarly, the set of positive rational numbers, $\mathbb{Q}^+$ is an extension of the positive natural numbers $\mathbb{N}^+=\{1, 2, \cdots\},$ which allows us to solve equations of the form $ax=b.$
Following this pattern, I considered solving exponential equations of the form $x^a=b$ over $\mathbb{N}^+.$ Such equations can be solved in the quotient set of $\mathbb{N}^+\times\mathbb{N}^+$ by the equivalence relation (assuming the cancellation law for exponents) $$(a, b)\sim (c, d) \iff b^c=d^a.$$ Let me call this new set "the radicals of positive integers." We can extend the multiplication on $\mathbb{N}^+$ to radicals by setting $$[(a, b)].[(c, d)]=[(ac, b^cd^a)].$$ Under multiplication
- $[(1, 1)]$ is an identity element
- Commutative and associative
Therefore, there is a possibility that we can extend this to a multiplicative group (Do we need the cancellation property too?).
However, radicals are not closed under addition and exponentiation. In that regard, this is not a nice extension.
My question is, how can we extend "the radicals of positive integers" to a larger set with better algebraic properties? Any answer in this direction is welcome.
