Consider the preorder defined by the Hasse diagram \(\boxed{no \rightarrow maybe \rightarrow yes}\)
Consider a potential monoidal structure with \(yes\) as the unit and \(min\) as the product.
Fill out a reasonable definition of \(min\) and check that it satisfies the conditions.
| \(min\) | no | maybe | yes |
|---|---|---|---|
| no | no | no | no |
| maybe | no | maybe | maybe |
| yes | no | maybe | yes |
Monotonicity: \(x_1 \leq y_1 \land x_2 \leq y_2 \implies x_1x_2 \leq y_1y_2\)
Suppose without loss of generality that \(x_1\leq x_2\)
then \(x_1x_2=x_1\) and \(y_1y_2 = y_1\) or \(y_2\)
In the first case, we know this is true because we assumed \(x_1 \leq y_1\)
In the second case, we know it from transitivity: \(x_1 \leq x_2\leq y_2\)
Unitality: \(min(x,yes)=x\)
Associativity: probably
Symmetry: table is symmetric.