I want to mention one thing, to soothe any puzzlement you may be feeling about the phrase "commutative monoidal preorder". In [Lecture 22](https://forum.azimuthproject.org/discussion/2084/lecture-22-chapter-2-symmetric-monoidal-preorders/p1), I started by defining "symmetric monoidal preorders". These are monoidal preorders where for any elements \\(x\\) and \\(y\\) we have

\[ x \otimes y \le y \otimes x \textrm{ and } y \otimes x \le x \otimes y .\]

In a preorder, two elements \\(a\\) and \\(b\\) are said to be **isomorphic** if \\(a \le b\\) and \\(b \le a\\), and we write this as \\(a \cong b\\). So in a symmetric monoidal preorder we have

\[ x \otimes y \cong y \otimes x .\]

But a _poset_ is exactly the same as a preorder where any pair of isomorphic elements are equal! So, a symmetric monoidal _poset_ is actually **commutative**:

\[ x \otimes y = y \otimes x . \]

I listed a bunch of "commutative monoidal posets" in [Lecture 23](https://forum.azimuthproject.org/discussion/2086/lecture-23-chapter-2-commutative-monoidal-posets/p1). As we saw, they're very common and important.

But in today's lecture we saw a method of building a bunch of monoidal preorders that are not posets but are still commutative. We can get one by taking the commutative monoid \\(\mathbb{N}[S]\\) and asserting any desired set of inequalities... along with all the other inequalities you can derive using the 6 rules I listed

That's why today I was talking about "commutative monoidal preorders". A simple example would be to take our set of things to be

\[ S = \\{ \textrm{H}, \textrm{O}, \textrm{H}_2\textrm{O} \\} \]

and put in the inequalities

\[ 2 \textrm{H} + \textrm{O} \le \textrm{H}_2\textrm{O} \]

\[ \textrm{H}_2\textrm{O} \le 2 \textrm{H} + \textrm{O} \]

which describe a "reversible reaction" turning hydrogen and oxygen into water _but also vice versa_. As we saw in the discussion of [Puzzle 70](https://forum.azimuthproject.org/discussion/2084/lecture-22-chapter-2-symmetric-monoidal-preorders/p1), this gives a commutative monoidal preorder that's not a poset, since we have

\[ 2 \textrm{H} + \textrm{O} \cong \textrm{H}_2\textrm{O} \]

but not

\[ 2 \textrm{H} + \textrm{O} = \textrm{H}_2\textrm{O} . \]

On the other hand, today's lemon meringue pie example was actually a poset, since there were no reversible reactions. You can't combine a yolk and a white to make an egg, for example!

\[ x \otimes y \le y \otimes x \textrm{ and } y \otimes x \le x \otimes y .\]

In a preorder, two elements \\(a\\) and \\(b\\) are said to be **isomorphic** if \\(a \le b\\) and \\(b \le a\\), and we write this as \\(a \cong b\\). So in a symmetric monoidal preorder we have

\[ x \otimes y \cong y \otimes x .\]

But a _poset_ is exactly the same as a preorder where any pair of isomorphic elements are equal! So, a symmetric monoidal _poset_ is actually **commutative**:

\[ x \otimes y = y \otimes x . \]

I listed a bunch of "commutative monoidal posets" in [Lecture 23](https://forum.azimuthproject.org/discussion/2086/lecture-23-chapter-2-commutative-monoidal-posets/p1). As we saw, they're very common and important.

But in today's lecture we saw a method of building a bunch of monoidal preorders that are not posets but are still commutative. We can get one by taking the commutative monoid \\(\mathbb{N}[S]\\) and asserting any desired set of inequalities... along with all the other inequalities you can derive using the 6 rules I listed

That's why today I was talking about "commutative monoidal preorders". A simple example would be to take our set of things to be

\[ S = \\{ \textrm{H}, \textrm{O}, \textrm{H}_2\textrm{O} \\} \]

and put in the inequalities

\[ 2 \textrm{H} + \textrm{O} \le \textrm{H}_2\textrm{O} \]

\[ \textrm{H}_2\textrm{O} \le 2 \textrm{H} + \textrm{O} \]

which describe a "reversible reaction" turning hydrogen and oxygen into water _but also vice versa_. As we saw in the discussion of [Puzzle 70](https://forum.azimuthproject.org/discussion/2084/lecture-22-chapter-2-symmetric-monoidal-preorders/p1), this gives a commutative monoidal preorder that's not a poset, since we have

\[ 2 \textrm{H} + \textrm{O} \cong \textrm{H}_2\textrm{O} \]

but not

\[ 2 \textrm{H} + \textrm{O} = \textrm{H}_2\textrm{O} . \]

On the other hand, today's lemon meringue pie example was actually a poset, since there were no reversible reactions. You can't combine a yolk and a white to make an egg, for example!