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\\).

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

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\\).