Let $P$ be a poset, or partially ordered set. Let $\le$ denote the reflexive order on $P$, and $<$ the corresponding irreflexive order. Let the phrase "a maximal pair" in $P$ refer to an ordered pair $(u,v)$ of elements in $P$ such that
- $u>v$,
- $u$ is a maximal element of $P$, and
- $v$ is a maximal element of the subposet $P\setminus\{u\}$.
We say a subposet $P'$ of $P$ is obtained by elementary collapse from $P$, if $P' = P \setminus \{u,v\}$ for some maximal pair $(u,v)$ of $P$. We say $P$ is collapsible if there is a sequence $$ P = P_0 \supset P_1 \supset \cdots \supset P_{k-1} \supset P_k = \left\{\ast\right\} $$ of subposets of $P$, in which $P_i$ is obtained by elementary collapse from $P_{i-1}$ for $i=1,2,\ldots,k$.
My questions are:
- I needed to prepare by myself the terminology for maximal pairs, elementary collapse, and collapsiblity; is there any better, more common way to refer to them?
- I am working on proving collapsibility for a class of posets, conjecturally being the face posets of some convex polytopes. Are there any general ways to prove collapsibility of a poset, assuming or not assuming realizations as convex polytopes?
