@[Dan](https://forum.azimuthproject.org/discussion/comment/18398/#Comment_18398): Mathematicians often pun the algebraic structure and the underlying set when context allows. We like to think of \$$\mathcal{V}\$$ as primarily a set, but carrying extra structure, so we still say that \$$\mathcal{V}\$$ has elements, even though formally we might define it as a tuple of a set \$$V\$$ and some operators and relations.

For instance, when talking about \$$\mathbb{R}\$$ and \$$\mathbb{C}\$$, it's understood that we're referring to the "unique Dedekind-complete ordered field" and the unique algebraic extension of the former by \$$\sqrt{-1}\$$, respectively, even though we still write \$$x \in \mathbb{R}\$$. And the standard topology for the cartesian product \$$X \times Y\$$ of two topological spaces is the component-wise topology, even though the lexicographic topology is also completely valid -- so we still just say \$$(x, y) \in X \times Y\$$, leaving the structure implicit.

It starts to matter more when talking about homomorphisms between structures sharing the same underlying set. This is when you start to see things like \$$\ln : \langle \mathbb{R}, \times \rangle \to \langle \mathbb{R}, + \rangle\$$, where we want to be very explicit about how the morphism interacts with the structure.