This looks great, Matthew! Puzzle 199 would have been hard for many students because the "workhorses" required to solve this puzzle are the distributive laws:

\[

a \otimes \left( \bigvee\_{b\in B} b\right) = \bigvee\_{b \in B} (a \otimes b) \quad ; \quad \bigvee\_{b\in B} (b \otimes a) = \left(\bigvee\_{b \in B} b\right) \otimes a

\]

which hold for every element \\(a\\) and every subset \\(B\\) of a commutative quantale \\(\mathcal{V}\\). The problem is that these need to be proved starting with the definition:

**Definition.** A **quantale** \\(\mathcal{V}\\) is a closed monoidal poset that has all joins.

The solution, as you so deftly note, is that \\(\mathcal{V} \\\) being closed implies that the operation \\(a \otimes - \\) is a left adjoint, so it preserves all joins. And first distributive law above says precisely that \\(a \otimes - \\) preserves all joins!

The second one says that \\(- \otimes a\\) preserves all joins, but in a _commutative_ quantale we have \\(- \otimes a = a \otimes - \\).

Those who are puzzled can look back at [Lecture 16](https://forum.azimuthproject.org/discussion/2031/lecture-16-chapter-1-the-adjoint-functor-theorem-for-posets/p1), where we proved this:

**Theorem.** If a monotone function \\( f: X \to Y \\) is a left adjoint, it preserves joins whenever they exist. That is, whenever a set \\( S \subseteq X \\) has a join we have

\[ f (\bigvee S ) = \bigvee \\{ f(x) : \; x \in S\\} . \]

But the conditions of 'preserving joins' and 'being a left adjoint' are more closely connected that this theorem acknowledges. In [Lecture 16](https://forum.azimuthproject.org/discussion/2031/lecture-16-chapter-1-the-adjoint-functor-theorem-for-posets/p1) we went on to prove this:

**Adjoint Functor Theorem for Posets.** Suppose \\(X\\) is a poset that has all joins and \\( Y \\) is any poset. Then a monotone map \\(f : X \to Y \\) is a left adjoint if and only if it preserves all joins.

Using this, we see that if \\(\mathcal{V}\\) is a monoidal poset with all joins, and the operation \\(a \otimes -\\) preserves joins, it must be a left adjoint! So, we get this nice result, which is a special case of Proposition 2.70 in _[Seven Sketches](http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf)_:

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

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

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

So, to help out future students, I'll add some of this material to the lecture!

\[

a \otimes \left( \bigvee\_{b\in B} b\right) = \bigvee\_{b \in B} (a \otimes b) \quad ; \quad \bigvee\_{b\in B} (b \otimes a) = \left(\bigvee\_{b \in B} b\right) \otimes a

\]

which hold for every element \\(a\\) and every subset \\(B\\) of a commutative quantale \\(\mathcal{V}\\). The problem is that these need to be proved starting with the definition:

**Definition.** A **quantale** \\(\mathcal{V}\\) is a closed monoidal poset that has all joins.

The solution, as you so deftly note, is that \\(\mathcal{V} \\\) being closed implies that the operation \\(a \otimes - \\) is a left adjoint, so it preserves all joins. And first distributive law above says precisely that \\(a \otimes - \\) preserves all joins!

The second one says that \\(- \otimes a\\) preserves all joins, but in a _commutative_ quantale we have \\(- \otimes a = a \otimes - \\).

Those who are puzzled can look back at [Lecture 16](https://forum.azimuthproject.org/discussion/2031/lecture-16-chapter-1-the-adjoint-functor-theorem-for-posets/p1), where we proved this:

**Theorem.** If a monotone function \\( f: X \to Y \\) is a left adjoint, it preserves joins whenever they exist. That is, whenever a set \\( S \subseteq X \\) has a join we have

\[ f (\bigvee S ) = \bigvee \\{ f(x) : \; x \in S\\} . \]

But the conditions of 'preserving joins' and 'being a left adjoint' are more closely connected that this theorem acknowledges. In [Lecture 16](https://forum.azimuthproject.org/discussion/2031/lecture-16-chapter-1-the-adjoint-functor-theorem-for-posets/p1) we went on to prove this:

**Adjoint Functor Theorem for Posets.** Suppose \\(X\\) is a poset that has all joins and \\( Y \\) is any poset. Then a monotone map \\(f : X \to Y \\) is a left adjoint if and only if it preserves all joins.

Using this, we see that if \\(\mathcal{V}\\) is a monoidal poset with all joins, and the operation \\(a \otimes -\\) preserves joins, it must be a left adjoint! So, we get this nice result, which is a special case of Proposition 2.70 in _[Seven Sketches](http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf)_:

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

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

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

So, to help out future students, I'll add some of this material to the lecture!