Daniel Wang wrote:

> $$
\text{monoid} & \text{effect of b) on $\mathcal{X}(x,y)$} \\\\
(\oplus, \tt{false}) & \text{If $\mathcal{X}(x,y)$, then for all objects $z$, either $\mathcal{X}(x,z)$ or $\mathcal{X}(z,x)$ or both} \\\\
(\wedge, \tt{true}) & \text{If $\mathcal{X}(x,y)$ and $\mathcal{X}(y,z)$, then $\mathcal{X}(x,z)$} \\\\
(\leftrightarrow, \tt{true}) & \text{If $\mathcal{X}(x,y) = \mathcal{X}(y,z)$, then $\mathcal{X}(x,z)$} \\\\
(\vee, \tt{false}) & \text{Either $\mathcal{X}(x,y) = \tt{false}$, or $\mathcal{X}(x,y) = \tt{true}$, for all objects $x,y$.} \\\\

Excellent, and good use of LaTeX too!

In [Lecture 30](https://forum.azimuthproject.org/discussion/2124/lecture-30-chapter-1-preorders-as-enriched-categories) I consider the second row in detail: with this particular monoidal structure on \\(\mathbf{Bool}\\), a \\(\mathbf{Bool}\\)-enriched category is just a preorder, since

If \\(\mathcal{X}(x,y) = \mathcal{X}(y,z)\\), then \\(\mathcal{X}(x,z)\\)

is the good old transitive law. Let me think a bit about the other three: I haven't really studied them, but they must be good for something!