Let $F$ be the set of functions over real numbers. Consider the structure $M = \langle F, <, \leq, =, \geq, > \rangle$ where the $<, \leq, =, \geq, >$ are defined as asymptotic notions $o$, $O$, $\Theta$, $\Omega$, $\omega$. For example, for $f,g\in F$, $f\leq g$ iff $f \in O(g)$.
Consider the first-order theory of this structure $T = Th(M)$.
Is $T$ finitely axiomatizable?
Does $T$ have quantifier elimination property?
Is $T$ a decidable theory?
Is there a references discussing logical and model theoretic properties of this theory (statements similar to those above)?