URL pointed out in a different forum, that the various products have been studied under the names of join and incomplete join, citing, for example DMY Sommerville (1929). These were further studied in the seventies. However, these do not constitute a discovery of new products.

It is fairly easy to place points at the dual of prisms, and throw a convex hull over these. These points exist. But then to suppose that the result is a product is not discussed. The strong relation between the regular solids and the products is one of the points of my approach. Yet the science is not advanced enough to make the connection between the products and the regular solids in Coxeter (1946, 1971), the 1946 book considered the semenial reference in the subject.

The year 2002 was the first year that I had internet connection beyond downloading. This was when I joined Mangus Wenninger's polytope mailing list, to which the likes of John Conway, Norman Johnson, Richard Klitzing, Jonathon Bowers, Guy Inchbald, and other polytope enthusists (professional and ameteur), were members I introduced a number of new concepts and notations to the list, many of these being novel advancements.

The tegum product was introduced as a proclaimation that the dual of the prism shall be called a tegum, and its general discription followed. Antigegum was introduced at this time too. Various ill-recalled conversations took place, but essentially, it amounts to that until this time, the dual of a prism was a join, covered by a blanket. Norman Johnson said exactly that, including that the join was a sum, since it followed the p(XY)=p(X)+p(Y) where p() is the vertex-count

The meaning of product given was that some property p(x) exists where p(XY)=p(X).p(Y) is an algebraic calculation and that p(XY)=p(X)+p(Y) is a sum, resulting in an algebraic sum. Given that the join rarely consists of more than a blanket-show of points, and explicitly stated as a sum, that the product represents new thought.

So we come to that the various products consist of that over the surtope consist, pre-pended with content and/or appended with the nulloid. Appending a '1' at the right-hand side, will cause the next column to add, and this leads to what has been seen as a sum. This is why the face-count of a prism is the sum of face-counts, because the prism-function pre-pends a '1'.

The role of the nullitope and set theory was discussed as well. The previous model that the nulloid represents the empty set is clearly false, since it is the down-incidence of a polytope, and it is not shared with other polytopes. Specifically, the point of down-incidence represents the grouping of elements that are and are not part of the polytope. It is instead, the 'name' or 'identity' of the polytope, as much as a name represents an individual, but is no presence on the individual.

The set-theory model falls to bits when the notion of 'union' is discussed. This has no meaning in polytopes (it gives a multitope), and the up-incidences is a different polytope where the surtope represents the down incidence. Norman Johnson worked this idea into 'polytope clusters'.

The metric products of the various polytope products were never discussed outside of my hest. The various radiant products (tegum, prism, later crind) are coherent, in that measures against the powers of unit-lengths, represent a coherent scale of units against L^n, and that these units stand in the ratio of 1:n! for tegum:prism units. The crind-units represent units of sphere.

Given the request to review this by V Hernández, we come to the conclusion that that while the join name was used to describe arrangements of points of tegum and pyramid products, it never went past this and that as late as this century, mainstream science understood it to represent a sum, rather than a product. Further noted is that the pyramid product is an 'incomplete join', although this is the product of the full surtope consist including the units at both ends.