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 == 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