MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there a standard name for order-preserving maps $f\colon P\to Q$ of posets with the property that the image of a lower set is a lower set, or equivalently if $q\leq f(p)$ then there exists $p'\leq p$ with $f(p')=q$?

If you view the poset as a category, then this condition says that the functor associated to $P$ is surjective on in-stars.

In my research I need to consider the category of posets with these kinds of morphisms and I would like to know their name. For example, if $P$ and $Q$ are face posets of regular CW complexes, then this property says that the image of each closed cell is a closed cell.

share|cite|improve this question
You don't also insist that the map is order-preserving? – Joel David Hamkins Jul 19 '13 at 15:40
By maps of posets I meant order preserving. I'll edit. – Benjamin Steinberg Jul 19 '13 at 15:47
I suppose it could also be ambiguous whether one means $p\leq q\iff f(p)\leq f(q)$, or $p\leq q\to f(p)\leq f(q)$ or $p\lt q\to f(p)\lt f(q)$. – Joel David Hamkins Jul 19 '13 at 16:19
@JoelDavidHamkins, by order preserving I mean $a\leq b$ implies $f(a)\leq f(b)$. – Benjamin Steinberg Jul 19 '13 at 16:37
@The Masked Avenger. Now that you mentioned it, it is actually a topological condition. If we give posets the topology where the lower sets are precisely the open sets, then the order preserving maps are precisely the continuous functions, and the open maps are precisely the maps that map lower sets to lower sets. Similarly, if we give each poset the topology where the lower sets are precisely the closed sets, then the closed maps are precisely the maps that map lower sets to lower sets. – Joseph Van Name Jul 19 '13 at 21:22
up vote 4 down vote accepted

I think one name for this is a simulation.

share|cite|improve this answer
thanks for this. I will wait to see if poset people have a different name before accepting. – Benjamin Steinberg Jul 20 '13 at 5:47

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.