Yigor, are you a mathematician? Your way of explaining the Logic behind the words, made me wonder .D
Anyways, an Position is a noun which describes either the physical or psychological state of something. But never could describe, the process to move from position A to position B. which means Transposition cannot describe the transposing (!!!!) from position A to the Transposition B from the usual position C. A=default position =/= chess starting position C=Our relating Position and B=our Position which is the same as our related position C but reached via another move order
What do I do with chess if I don't play it ?!? 
U thought that 2 openings are necessary in order to make a transposition. However, it's not the case in general. U can take just 1 opening and alter the standard move order. 
Yigor, what you're saying would make sense if you took position A and then moved pieces (not making chess moves just changing the position) in order to arrive at position B. This would not be a transposition, it would just be one position being changed into another. When people use the word "transposition" to describe something like a game beginning as a Caro-Kann but then reaching a Queen's Gambit position or something like that what they mean is that a Queen's Gambit position has been arrived at via transposition from the Caro-Kann. E.g. what we are looking at momentarily is a position from the Queen's Gambit that can be continued in that opening but the moves order is not a Queen's Gambit move order, 2. d4 was played instead of 1. d4 etc... If you want to be strict about the meaning of the word in that sense the "transposing" we are talking about the permutation of the moves and the move order of the Queen's Gambit not something the process of changing from one opening to another. As Deidreskye has pointed out, however, the use of transposition to go from one opening to another is pretty much always the way it comes up in the literature as it is useful. You can also transpose to known positions from garbage move orders, but there's generally no point to talking about those. I have, however, sometimes seen very old games analyzed and a few of the first moves explained by annotator with something like "well chess theory then was not what is was now; now we would play ___ first" followed by at point in the game something like "but already here we have transposed into a Slav which is still played today." Perhaps Deirdreskye does not like it when annotators do that, or when they explain that one game (not opening or position) transposes into another game but I feel those are valid uses of the term. If you think about it the way you are thinking about it instead, as transposition being only the process of changing from one opening to another, then I guess you can imagine some abuse of language is happening to get that word to mean that thing; that's not really happening, though. If you look at it the other way, you will notice transposition (even by your mathematical definition) is occurring in order to arrive at your position B.
Hey, bro, thanks for this long explanatory post!
I started this (a bit humoristic) thread against DeirdreSkye who attacked an other participant saying that he doesn't know what is transposition. However, it turned out that Skye doesn't know the formal definition himself (well, now he has finally got it). I agree that this formal definition in mathematical terms of permutations of move orders is not terribly useful. It was just to show to our wonderful participant Skye that he's not as omniscient as he pretends to be. 
