@[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.

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.