Edit: My reading comprehension is low and I didn't notice that a quantale needs to be closed :)


I'm confused about the definition of quantale. You said:

**Definition.** A **quantale** is a closed monoidal poset \\( \mathcal{V}\\) that has all joins: that is, every subset of \\( S\subseteq \mathcal{V}\\) has a least upper bound \\(\bigvee S\\).

And then we sketched a proof of the following:

**Theorem.** If \\(\mathcal{V}\\) is a monoidal poset with all joins, \\(\mathcal{V}\\) is a quantale if and only if

\[ a \otimes \left( \bigvee\_{b\in B} b\right) = \bigvee\_{b \in A} (a \otimes b) \]

for every element \\(a\\) and every subset \\(B\\) of \\(\mathcal{V}\\).

But I thought that a quantale is a monoidal poset with all joins. Why do we need prove this extra condition about the monoidal product commuting with joins?