I'm trying to study posets as part of a larger algorithm I'm creating. At the advice of a colleague, I'm reading Enumerative Combinatorics by Richard Stanley, but despite the length of the chapter, he doesn't seem to answer my fundamental question:
What is the name of the maximal chain that contains one element from every level (so all elements in $P_S$ are either in the chain or on the same level as an element on the chain)? Or to put it more technically:
Given a finite poset P with underlying set $P_S$, define Whatever this structure is called $G$ as follows:
$\forall e \in P_S: e \in G \vee \exists k \in G : k \not\le e$
I hope to identify this structure so I can better understand its properties and use it effectively in my work.
You could foliate the poset by using antichains, then you would have a concrete notion of a layer. You can then pick out elements layer by layer and form a chain. You can read more about this here https://en.wikipedia.org/wiki/Mirsky%27s_theorem