Consider some partially ordered set $(E,\leq)$. Assume either that it is countable with the discrete topology, or that it has some topology compatible with the order, preferably one that makes it into a Polish space. Now consider the semigroup of (continuous) order-preserving functions from $E$ to $E$. What is known about the structure of this thing?

That is the most general set-up. The most specific examples I'm interested in are $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{N}^n$, and maybe $\mathbb{R}^n$. So some restrictions which could also be interesting if they have better results:

- Assume it is totally ordered, not just partially ordered
- Maybe even well-ordered
- It could have a minimal element, which we require to be mapped to itself
- Perhaps restrict to the functions which in some suitable sense go to infinity
- The case where we require them to be
*strictly*increasing is less interesting, but if there are any results there I'm all ears

The reason I care about these semigroups is because I want to consider probability measures and random walks on them, which is why I had the restriction of it having a nice topology. This also means the ideal result would look something like "all subsemigroups are [explicit description], they're closed and completely simple if [explicit condition]" -- or any other classification of its subsemigroups of various types.