Non-algebraic structures?

Question

We call group, ring, field,... "algebraic structures". Do we have similar analogue for transcendental numbers? If not, then how do we study interactions between various transcendental numbers?

Also, both irrational numbers and transcendental numbers are uncountable, is it a reason for their inability to form algebraic structures?

Answer 1

Your question doesn't really make sense. The term "algebraic numbers" is only very loosely related to the term "algebraic structure": both have to do with algebraic operations, but there is no reason at all that an algebraic structure needs to consist of algebraic numbers (or numbers of any kind, for that matter).

In particular, there is nothing from stopping you from using the usual structures of abstract algebra to study relationships between different transcendental numbers. For instance, $\mathbb{C}$ is a field which contains all of the transcendental numbers. So you can study algebraic relationships between different transcendental numbers inside the field $\mathbb{C}$.

What is true is that (for instance) the set of all transcendental numbers does not form a field under the usual operations, since it is not closed under those operations (for instance, $\pi$ and $-\pi$ are transcendental, but their sum $0$ is not). However, you can still study them perfectly well by considering them as a subset of a larger algebraic structure.

Answer 2

In group theory there is the Frattini subgroup of a group $G$, which consists of all the elements of $G$ which do not belong to any minimal generating set. It is in some sense the collection of all the "least nice" elements of the group, yet it happens to be a (sub)group itself.

More generally, there is the universal algebra concept of a Frattini subalgebra of an algebra $\mathbf{A}$, which is the subalgebra generated by all elements of $A$ which do not belong to any minimal generating set for $\mathbf{A}$. You might find it instructive to try to determine, for example, the Frattini subgroup of the reals under addition, since this is both "bad" in the sense you seem to be asking for and also has "natural" structure.