Improvements to Article Minimal and Maximal are not the best terms as they do NOT imply uniqueness An order can have many minimal maximal elements ARP uses Least and Greatest which admittedly are a little less aesthetic but are consistent with standard Partial Order Theory I have made the Polytopes as Posets section both much clearer and more rigorous and moved the incidence definition to the Incidence Matrix section I have made the introduction more readable to non experts by deferring i e moving the more advanced concepts further down the article Nothing has been deleted I strongly feel that my 5 quads picture should be at the top because it immediately and instantaneously gets the concept across before the more difficult task of understanding the text at least to anyone capable of understanding the text It is much less difficult task to scroll up from the next section that refers to it It also I think makes the article more appealing at first glance which I think matters There are inaccurate statements in the Realizations section Four vertices do not make a square this also requires edges at least The intended meaning is clear anyway of course I will probably fix this at some stage unless one of you cares to first The problem for me is the lack of simple non abstract defn of polytope The intro paragraph is badly worded It first says that an abstract polytope is a structure that captures the combinatorial structure of a classical polytope and then contradicts itself by saying that an abstract polytope encompasses more than just classical polytopes It really should state at the outset that an abstract polytope is a generalization of a classical polytope and is more inclusive than a classical polytope Also please try to avoid textbook style wording and tone such as using we as much as possible try to word it as an encyclopedic description of the subject rather than a pedagogical introduction to the subject See WP style I hope I m not coming across as being nitpicky I m just trying to improve the article as you are P S see also WP NOT TEXTBOOK Ta for feedback points taken and your diplomatic remarks much appreciated I was aware of the contradiction But keep in mind that the introduction is an introduction Here simplicity and clarity should be the paramount objectives It is not a formal definition and clearly says so The second statement doesn t so much contradict as qualify the preceding one I have added informally speaking and in fact which I hope will meet your objection I can t see how to improve it further without losing reading clarity Hope this is okay for you now and do always feel free to comment bearing in mind that one sometimes ends up with egg on the face as I have a few times We shall try to avoid we Season s Greetings Regarding realizations the article now says that the pyramid we all met in primary school is a realization of the abstract polytope with thegive Hasse diagram This realization has infinitely many points But if I remember rightly the realization of a regular AP as defined in ARP is a mapping from the set of vertices of the AP onto some Euclidean space so it can have only as many points as the AP has vertices Again if I remember correctly ARP says something like this might not be a satisfactory definition to everyone but it s the one we are working with here I may have put my foot into my mouth here but this means that if I remember correctly the article is currently incorrect regarding realizations I am currently musing over precisely what Realization means However I am not sure your objection is correct It seems that a realization of an AP is a traditional pol such that the face posets of both are isomorphic ie a bijective order preserving map exists It is not required that all the internal points of a trad pol be mapped I think we only have to worry about the vertices of the tr