Sections of an abstract polytope

Question

An abstract polytope is a certain kind of partially-ordered set. Its elements or "faces" are ranked by "dimension" and also partially ordered via a pairwise "incidence" relation between elements of adjacent ranks.

For some abstract polytope $P$, of which $F$ and $G$ are faces such that $G \le F$, the set of faces $H$, such that $G \le H \le F$, is a section of $P$ and is written $F/G$.

I define a sub-polytope of $P$ as a subset of $P$ which is also an abstract polytope.

What is the relationship between sections and sub-polytopes?

• Are all sections of $P$ necessarily (sub-)polytopes in their own right?
• Are all sub-polytopes of $P$ necessarily sections?

Answer 1

Any polytope $P$ has a single unique nullity $Ø≤P$.

Any subpolytope or subelement $F$ then necessarily requires $Ø≤F≤P$ and yes, both the polytope itself as also that subpolytope can be identified with the according sections: $$P\cong P/Ø,\ F\cong F/Ø$$ therefore all subpolytopes indeed are sections.

Whereas when $Ø<G≤F≤P$ then $F/G$ certanly is a section, but not a subpolytope (subelement) of $P\cong P/Ø$.

None the less, $P/G$ happens to be the $G$-figure of $P$. Thence any $F/G$ can be considered a polytope on its own right for sure. In fact it just represents the subdiagram of the Hasse diagram of $P$ (ie spanned between $Ø$ and $P$) which is spanned between $G$ and $F$. And that one for sure is a Hasse diagram itself.

--- rk

Answer 2

Every section of a polytope is a polytope. See for example:

• McMullen & Schulte (2002). Abstract Regular Polytopes, CUP, Page 23.
• Ilya Scheidwasser (2017). "Contractions of Polygons in Abstract Polytopes, Part I", Contributions to Discrete mathematics, Volume 11, Number 2, Pages 19–42. ISSN 1715-0868

As the question defines a sub-polytope, a section of $P$ is therefore indeed a sub-polytope of $P$.

I can find no direct statement going the other way, however it seems to follow easily enough:

Every sub-polytope has a maximal and a minimal element. Every connected max/min pair $G \le F$ has a section $F/G$ which is a polytope.

It is more or less by definition that no sub-polytope of the same rank (i.e. with the same max/min pair) as the original can exist. Therefore the section $F/G$ must be the unique polytope with max/min pair $G \le F$.

Sections exist for every possible pair $G \le F$. They are the only sub-polytopes for each pair. No other sub-polytopes can therefore exist. The set of sections of a polytope is therefore also the full set of sub-polytopes.

So I conclude that the sections of $P$ are just its sub-polytopes.