# Semigroups of nondecreasing functions

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:

1. Assume it is totally ordered, not just partially ordered
2. Maybe even well-ordered
3. It could have a minimal element, which we require to be mapped to itself
4. Perhaps restrict to the functions which in some suitable sense go to infinity
5. 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.

• The semigroup of all order preserving maps on 1,...,n is already fairly complicated. – Benjamin Steinberg Sep 21 '17 at 23:55