" For more, google on for example polytope subface. — Cheers,    

 In the absence of any dissent, and in the interests of consistency and clarity, I shall implement this very soon. In any case, it would be better than the present ambiguity, and we can still continue to haggle over terminlogy afterwards. 

 This will affect our oft-updated flag defn! But I shall not flag in my quest for mathematical perfection.     

 : Done - together with some readability improvements. A much clearer exposition now, I think.    

 ::I am concerned that includes is no improvement on contains. Informally it still has connotations of one element inside another, and more strictly if ab represents an edge element then in general vertex element a is not a subset of ab (only φ and ab are) and therefore is not included in ab. Also, from what you say I assume that inclusion is not used in ARP. Perhaps it might be best to simply use the symbol ""<"" on these occasions, as ARP does. — Cheers,    

 ::: If we formalise completely, then our faces are sets of vertices; eg the vertex a is the set {a}, the edge ab is the set {a,b}. With this definition, it is true that a is a subset of ab, or ab includes a. Of course, we can't formalise all the time - it probably takes 50 lines to prove in arithmetic that a + b = b + a. The purpose of formalising is just to be sure we can be precise when necessary.  So in practice we always abbreviate {a, b} to ab. Is contained in is definitely inconsistent, because it means ""is an element of"", so includes is an improvement, and I swapped over precisely because of your correct observation that we had been using contain ambiguously. 

 ::::This hark back to the ""element abc of some polytope P is not {φ, a, b, c, ab, bc, ca, abc}"" issue. Likewise there is no a priori need for abc to be the set {φ, a, b, c}. In general abc is merely an element and is not per se a set. The ""abc is a set"" idea is just one interpretation of the abstract structure; it can be a useful simplification in combinatorics - which is where AP theory began - but otherwise it blurs the distinction between the element and the associated subsets of which abc is also an element. This is key to Johnson's formalism, which he says is entirely equivalent to ARP, so we need to respect it. To paraphrase your comment, if we formalise completely then our faces are mere elements and NOT sets of vertices; eg the vertex a is NOT the set {a}, the edge ab is NOT the set {a,b}, and it is NOT true that a is a subset of ab. — Cheers,    

 :::::1. You are right in saying that in AP theory, it is not required that faces are defined as vertex sets; an AP can be completely formalised as a partial order of elements (faces) which are not further divisible. Therefore, my argument in favour of the ""includes"" term is weakened (tho' not destroyed).  

 :::::2. However, you yourself, if I recollect correctly, were confused/dissatisfied, as I should have been, by the previously ambiguous use of ""Contains"". Flags are sets of faces, and do therefore contain faces (as elements). If we also use ""contains"" in the sense edge ab contains vertex a, we are now using it to mean ""<"" , i.e. our partial order relation. It is definitely NOT a good idea within the same project to use the same term with different mathematical meanings. I have stated that we should use the symbol most of the time. But we also want a readable article.  For example, in the article I used the term inclusion pairs which I think reads very nicely.  I really can't think of a better term. ""Belongs to"" rather implies set membership. ""Is a subface of"" doesn't appeal to me. I used to use the terms ""is [an] inferior/superior of/to"" - so we could use those. (I'm too chicken to use ""pecks"", and Alice might revert ""jabberwocks""). Any other candidates, anyone? 

 :::::3. Hmm... you could actually formalise in such a way that, eg, a ∈ ab ∈ abc, by defining each face as the set of its subfaces plus its name symbol. To do this, define φ, a, ab, abc respectively as φ, {φ, a}, {φ, a, b, ab} and {φ, a, b, c, ab, ac, bc,