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

Hi,

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?