A poset that's the union of the lower sets

Let $(P,\leq)$ be a poset, and let $\downarrow\! p = \{ x\leq p\}\subseteq P$. Let $M\subseteq P$ be the subset of all maximal elements of $P$.

Question: is there a specific term for a poset $(P,\leq)$ such that $P \subseteq \cup_{m\in M} \downarrow\! m$? That is, $P$ is equal to the union of the lower sets of all maximal elements?

If it makes any difference, in my particular case I have that $P = L \setminus \{ \sup L\}$ where $L$ is a complete lattice.

-
If I understood your question correctly, in the special case you're working with, you're just saying that the lattice $L$ is coatomic. – Martin Sleziak Jan 11 '12 at 16:14
@Martin: perfect. If you post that as an answer I will accept it (and I can live with reversing my [at present arbitrary choice of] order and make it atomic instead :p). – Willie Wong Jan 11 '12 at 16:19
Only now I realized, that this is (in ZFC) equivalent to (dual of) the condition from Zorn's lemma. (Every chain has a lower bound.) I hope I am not missing something.\\ Since ZL is used so frequently, one would expect for that condition to have a name, but I do not remember seeing it named anywhere. – Martin Sleziak Apr 1 '14 at 12:03
@MartinSleziak: to be honest, it has been so long since when I asked the question I can't remember exactly why I asked, and whether ZL was anywhere in the original motivation where I came across sets of this form. – Willie Wong Apr 1 '14 at 13:57
Sorry, I made a mistake in my previous comment. It is true that in an inductive poset there exists a maximal element above each element. (For example, see Arturo Magidin's answer to this question.) But these two conditions are clearly not equivalent. – Martin Sleziak Apr 2 '14 at 16:13

If we have a poset $(P,\le)$ then we can adjoin a new greatest element by setting $\overline P=P\cup\{1\}$ and $p\le 1$ for each $p\in P$. As OP mentioned in his post, this is the situation he is working with. (In this case $\overline P$ is the complete lattice $L$.)

In the above situation:

• $m$ is a maximal element of $P$ if and only if $m$ is a coatom in $\overline P$;
• the condition $P \subseteq \cup_{m\in M} \downarrow\! m$ means that every element of $P$ has a coatom of $\overline P$ above it (this is equivalent to: every element of $P$ has a $P$-maximal element above it).

So together we get that $P$ has the required property if and only if $\overline P$ is coatomic.

The notions of coatom and coatomic poset are dual to the notions of atom and atomic poset. The latter seem to be used more frequently.

Based on a guess what phrase could be used if someone would define such thing as described in the question, I browsed a little through the results of searches similar to poset "minimal element below". The only thing I found out was that in study of preference relations the name smooth relation is used sometimes, e.g. Coherent systems by Karl Schlechta p.79:

A strong requirement for the relation, which we find difficult to justify intuitively as a relation property, is smoothness. Essentially, it says that elements are either minimal, or there is a minimal element below them.

Although I do not think this particular case is of much interest for the OP.

-