How can I represent by generators and relations the set of real numbers

Question

How can I represent $\mathbb{R^{+}}$,$\mathbb{R^{*}}$,$\mathbb{Q^{*}}$,$\mathbb{Q^{+}}$ by generators and relations on them? Like that $\langle a,b,c,\cdots\mid \cdots\rangle$?

Answer 1

I guess that $\mathbb{R}^+$ and $\mathbb{Q}^+$ denote the additive groups, whereas $\mathbb{R}^*$ and $\mathbb{Q}^*$ denote the multiplicative groups of nonzero elements.

Note that $\mathbb{Q}^*$ is isomorphic to the product of $\{1,-1\}$ with a free group on a countable basis, so its presentation should be clear.

Note also that $\mathbb{R}^*\cong\{1,-1\}\times \mathbb{R}^+$ via the map $x\mapsto(\operatorname{sgn}(x),\log\lvert|x\rvert)$, so you just need a presentation of $\mathbb{R}^+$, which is isomorphic to the direct sum of $2^{\aleph_0}$ copies of $\mathbb{Q}^+$ (it needs choice, so the isomorphism cannot be explicitly written down).

Hence you're reduced to a presentation of $\mathbb{Q}^+$.

Now use the fact that $\mathbb{Q}^+$ is a torsion-free divisible abelian group.