Yes, **order-preserving** means the same as **monotonically increasing**, and the book uses **monotone** as a shorter synonym. So, in the book, a **monotone function** \$$f : A \to B\$$ between preorders \$$A\$$ and \$$B\$$ has

$$a \le a' \textrm{ implies } f(a) \le f(a')$$

for all \$$a,a' \in A\$$. Other people also talk about **order-reversing** or **monotonically decreasing** functions, that obey

$$a \le a' \textrm{ implies } f(a') \le f(a)$$

I forget how Fong and Spivak describe these, but every poset \$$A\$$ has an **opposite** poset \$$A^{\mathrm{op}}\$$ , in which the order relation is reversed, and a monotonically decreasing function from \$$A\$$ to \$$B\$$ is the same as a monotonically increasing function from \$$A^{\mathrm{op}}\$$ to \$$B\$$.