I'm a little confused about how the bowl renewable resource works in \\(\mathbb N [S]\\). I claim that if \\(x, x' \in \mathbb N [S]\\) and \\(x \leq x'\\) then the number of bowls in \\(x\\) and in \\(x'\\) are the same. This is because the relation \\(\leq\\) is built from applying the monoidal preorder rule ( \\(x \leq x', y \leq y' \implies x \otimes y \leq x' \otimes y'\\) ) to some set of "generating" relations. These generating relations are the reflexive relations along with:

\[ [\textrm{bowl}] + [\textrm{yolk}] + [\textrm{white}] + [\textrm{egg shell}] \le [\textrm{bowl}] + [\textrm{egg}]\]

Since all of the generating relations have the same number of bowls on both sides, so will any relation coming from applying the monoidal preorder rule.

This also matches up with my intuition, because every reaction in the kitchen starts and ends with the same number of bowls.

I bring this up because it leads me to a problem when trying to come up with a right adjoint. John suggests looking for:

> the biggest combination of bowls, eggs, yolks, whites and egg shells, say \\(x \in \mathbb{N}[S] \\), such that

> \[ f(x) \le b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}] . \]

Let \\[ r= [\textrm{bowl}] + b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}] + 0 [\textrm{egg shell}]\\] and \\[ r'= 2[\textrm{bowl}] + b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}] + 0 [\textrm{egg shell}].\\] Now \\(f(r) = f(r') = b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}] \\). So by the reflexive rule we have \\[f(r) \le b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}] , \quad f(r') \le b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}] .\\] This means that \\(g(b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}])\\) must be bigger than both \\(r\\) and \\(r'\\). But if my claim above holds, then there is no such element of \\(\mathbb N[S]\\).

I think this shows that there can be no right adjoint \\(g\\), but that doesn't feel quite right. Where have I gone wrong?

\[ [\textrm{bowl}] + [\textrm{yolk}] + [\textrm{white}] + [\textrm{egg shell}] \le [\textrm{bowl}] + [\textrm{egg}]\]

Since all of the generating relations have the same number of bowls on both sides, so will any relation coming from applying the monoidal preorder rule.

This also matches up with my intuition, because every reaction in the kitchen starts and ends with the same number of bowls.

I bring this up because it leads me to a problem when trying to come up with a right adjoint. John suggests looking for:

> the biggest combination of bowls, eggs, yolks, whites and egg shells, say \\(x \in \mathbb{N}[S] \\), such that

> \[ f(x) \le b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}] . \]

Let \\[ r= [\textrm{bowl}] + b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}] + 0 [\textrm{egg shell}]\\] and \\[ r'= 2[\textrm{bowl}] + b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}] + 0 [\textrm{egg shell}].\\] Now \\(f(r) = f(r') = b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}] \\). So by the reflexive rule we have \\[f(r) \le b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}] , \quad f(r') \le b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}] .\\] This means that \\(g(b [\textrm{egg}] + c[\textrm{yolk}] + d[\textrm{white}])\\) must be bigger than both \\(r\\) and \\(r'\\). But if my claim above holds, then there is no such element of \\(\mathbb N[S]\\).

I think this shows that there can be no right adjoint \\(g\\), but that doesn't feel quite right. Where have I gone wrong?