Can you make some sort of structure out of transcendental numbers?

Question

Let $T$ be the set of trancendental numbers over $\Bbb{Q}$. Then it is an easy proof that for all $a,b \in T$, either $a + b$ or $ab$ or both are transcendental. What if you defined the operation $*$ on $T$ to be $+$ where appropriate and $\cdot$ where appropriate. And if both are appropriate prefer $+$. Then $T$ is closed under $*$. Further $a*b = b*a$ since we're in a field. $a*b*c$ could underneath be $a+b c$, so this structure is non-associative. Is there a better way to build an algebraic structure from them?