Why is this not a category?

Question

Why need the composite of two monotone functions not be monotone?

This is from Rings and Categories of Modules, Anderson and Fuller, page 7.

Answer 1

Actually the claim is false: composition of monotone functions is monotone.

Clearly the composite of two order preserving maps is again an order preserving map. If $f \colon P \to Q$ is an monotone increasing map and $g \colon Q \to R$ is a monotone decreasing it is easy to see that $$x \leq y \Rightarrow f(x) \leq f(y) \Rightarrow g(f(x)) \geq g(f(y))$$ and so $g \circ f$ is a reverse order map. Similarly for $f \colon P \to Q$ monotone decreasing and $g \colon Q \to R$ increasing. Finally in the case where both $f \colon P \to Q$ and $g \colon Q \to R$ are monotone decreasing we have that $$x \leq y \Rightarrow f(x) \geq f(y) \Rightarrow g(f(x)) \leq g(f(y))$$ hence $g\circ f$ is monotone increasing.