# Thread: Transformations of a set that can't be non-decreasing under any well-order

1. ## Transformations of a set that can't be non-decreasing under any well-order

Let X be a set. Which functions from X to itself cannot be non-decreasing if we equip X with any well-order ≤, the same in X as the domain and X as the codomain?

Such functions obviously exist. For example, if we take X={0,1} and the function f(x)=~x ("~" denoting negation), then there is no well-order on X making this a non-decreasing function.

But what are these functions in general? Does a composition of such functions also have this property?

2. ## Re: Transformations of a set that can't be non-decreasing under any well-order

I've just realized that the last question is not the one I need an answer to. I wanted to ask whether a composition of functions that satisfy the negation of this property also does. That is, if $\displaystyle f,g:X\longrightarrow X$ are functions and there are two well-orders $\displaystyle \leq_f,\leq_g$ on $\displaystyle X$ such that $\displaystyle f : (X,\leq_f)\longrightarrow(X,\leq_f)$ and $\displaystyle g : (X,\leq_g)\longrightarrow(X,\leq_g)$ are both non decreasing, then is there a well-order $\displaystyle \leq_{f\circ g}$ on $\displaystyle X$ such that $\displaystyle f\circ g : (X,\leq_{f\circ g})\longrightarrow(X,\leq_{f\circ g})$ is non-decreasing?