Rationale behind tuple notation for structured sets

Question

Defining structured sets typically involves the convention of using a tuple of some sort; for example, the real line can be thought of as the quadruple $(\mathbf{R},+,\cdot,<)$. But this convention is often almost completely neglected post-definition. Why do mathematicians even bother?

Answer 1

This notation allows for unambiguous reference to the object. For instance $\mathbb{R}$ could refer to the group $(\mathbb{R},+)$ or to the ring $(\mathbb{R},+,\cdot)$. Usually either the distinction is clear from context, or it doesn't matter, and so the -tuple notation is dispensed with.

There can be situations where a set has, for instance, two different group structures on it, and then it comes in handy to distinguish which one is under consideration.

Answer 2

It is convenient to immediately make it clear that whatever structures set you define, is a set together with this and that extra things. You immediately know how many things to expect. Also, for pedagogical reasons, when first learning about, e.g., vector spaces, it is important to drive the fact that a vector space is a set together with extra structure. It is very common for students to ask (and, unfortunately sometimes to be asked) whether a given set is a vector space. The intention is to assume that set is equipped with the necessary extra structure that is somehow (magically?) obvious.

In any case, notation should be used sensibly and should not become a straitjacket. Once the reader get accustomed to whatever the object of study is, it becomes redundant to use the full tuple notation, unless ambiguity demands it.