Daniel Wang wrote:

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

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!