Exercise 2.8

The element \(e\) in the monoid \((M, \ast, e)\) serves as \(I\) in the discrete preorder \((\bf{Disc}_M, =, \ast, e)_{}\). The monoid multiplication \(\ast\) serves as the monoidal product \(\otimes\) and satisfies properties (b), (c), and (d) by definition of being a monoid. By virtue of being a discrete preorder, it satisfies (a) because \(x_1\) will always equal \(y_1\) and therefore with “=” as the order operator this condition comes down to \(x_1 \otimes x_2 = x_1 \otimes x_2\).