I was reading ring and categories of modules and I had a question.
The book is Frank W. Anderson and Kent R. Fuller’s Rings and Categories of Modules.
Let P be the class of all posets, $Hom\left(A,B\right)$ the set of all monotone maps (order preserving and order reversing ones), and $\circ$ the usual composition.
The author says the P is not a category because the composition of two monotone functions need not be monotone.
The following is my attempt.
Let $x\le y$:
If $f$ is an order preserving function and $g$ is an order preserving function, then $gf$ is also an order preserving function.
If $f$ is an order preserving function and $g$ is an order reversing function, then $gf$ is an order reversing function.
If $f$ is an order reversing function and $g$ is an order preserving function, then $gf$ is an order reversing function.
If $f$ is an order reversing function and $g$ is an order reversing function, then $gf$ is an order preserving function.
Did I get it wrong?
Thanks!
let $g=1/x,\;0<x\leq \frac{\pi}{2}$. $g$ is order reversing.
let $f=sin(x),\; 0<x\leq \frac{\pi}{2}$. $f$ is order preserving. composition: $fg=f(g(x))=sin(\frac{1}{x})$ is not order preserving nor reversing as we approach 0.