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:: Thank you, [[User_talk:Quondum|''Quondum'']], for adding that definition to the article. --[[User:DavidCary|DavidCary]] ([[User talk:DavidCary|talk]]) 19:57, 8 June 2014 (UTC)
== Algorithm? ==
I can't find description of algorithm that would match the one given in this article and used in example.
References define that an m-digit sequence <math>d_md_{m-1}d_{m-2}\ldots d_1</math> with a check digit <math>d_0</math> shall satisfy
<math>((\ldots((d_m*d_{m-1})*d_{m-2})*\ldots)*d_1)*d_0 = 0</math> where <math>d_m,\ldots,d_0 \in Q</math> and <math>(Q,*)</math> is a totally anti-symmetric quasigroup.
The step-by-step description of algorithm in this article uses interim digit with fixed initial value of 0 which means that
a slightly different equation is used, i.e. <math>((\ldots((0*d_m)*d_{m-1})*\ldots)*d_1)*d_0 = 0</math>.
I don't see any problem with this itself, because it's the same as computing with tacitly leading zero prefix, so all the properties should still hold.
However, the example then uses a quasigroup which '''not''' totally anti-symmetric, because, for example 2 * 5 = 5 * 2. It is only ''weak'' totally anti-symmetric. (As a side note I think this name is misleading because )
With weak totally anti-symmetric quasigroup, check digit calculated using the original formula as given in references would fail to detect adjacent transpositions, e.g. both 257 and 527 would have a correct check digit.
The scheme, nevertheless, appears to work exactly because of the fixed initial 0.
The recent [https://en.wikipedia.org/w/index.php?title=Damm_algorithm&diff=prev&oldid=614858206 edit] added possible justification for this stating that for this modified equation only a weak totally anti-symmetric quasigroup with additional property of {{math|1=''x'' ∗ ''x'' = 0}} is needed. I can easily see that other statements (definition of weak totally anti-symmetric, and that existence of weak of given order implies existence of totally anti-symmetric and that one can be constructed from another with appropriate permutation) are present in the referenced dissertation (and they are restated later in English in ScienceDirect article, referenced as well), but I can't find anything about using fixed prefix. Did I miss something or are we missing some important references? — [[User:MwGamera|mwgamera]] ([[User talk:MwGamera|talk]]) 19:11, 29 June 2014 (UTC)
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