Computational complexity theory considers functions $f : \mathbb{R}_{> 0} \rightarrow \mathbb{R}_{> 0}$ ordered where $f \leq g$ when there is a $x \in \mathbb{R}_{> 0}$ such that $f(x) / g(x) \leq C$. Or rather, we need to avoid division by $0$, so the condition is
$$f(x) \leq C g(x)$$
I am interested in the "group of complexity classes", constructed as follows:
Let $A$ be the abelian group of functions $f : \mathbb{N} \rightarrow \mathbb{R}_{> 0}$, where the group operation is pointwise product of functions. Let $N$ be the normal subgroup of functions $f$ such that there are $a,b \in \mathbb{R}$, and $n \in \mathbb{N}$, with $0 < a < 1< b$, such that $a < f(m) < b$ for $m \geq n$. Write $C = A/N$. $B$ is an abelian group, ordered where $[f] \leq [g]$ when there is $n \in \mathbb{N}$ and $C \in \mathbb{R}$ such that $f(m) \leq C g(m)$ for each $m \geq n$.
My question is, is there a structure theorem for this group?
Actually, I think the real group of interest is actually a certain totally ordered subgroup of this group. For instance, $[\text{sin}(x)]$ and $[\text{cos}(x)]$ are not related under this group.
Moreover, there is apparantly a second group law $\circ$, given by composition, so that e.g. $[\text{exp}(n)] \circ [\text{log}(n)] = [\text{id}]$. It would be nice if this group law distributed over multiplication.
So my actual (though admitedly more vague) question is, how might one pick out the totally ordered subgroup of interest in complexity theory, and does this have an interesting structure theorem?
There are very large natural classes of functions from reals to reals which are linearly ordered by domination (= eventual $>$). These yield linearly ordered subgroups of your original $A$ (and of $B$), and are best understood as substructures of very large ordered fields of "functions" (= formal series of certain types - e.g. Hahn fields).
For example, the set of "exponential polynomials" is even well-ordered with respect to domination - see e.g. the introduction to this paper of van den Dries and Levitz (although note that there appears to be a typo in the abstract).
In fact, in a sense the set of all functions you can build from polynomials, exponentiation, and logarithms is linearly ordered by domination - see this survey paper on transseries. (An even larger ordered field of "functions" is given by the (logarithmic) hyperseries.)