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:--[[User:W96|W96]] ([[User talk:W96|talk]]) 18:58, 18 January 2015 (UTC)
== In which cases does rearranging the rows not work? ==
Rearranging the '''rows of the body of the table''' in conjuntion with changing the entries correspondingly is done as follows:
Let e be the digit that shall be in the main diagonal (x*x=e for all x∈Q).
<br>(In the example is e=0.)
Then find the fixed point f of the column permutation in the column with the header e.
<br>(In the example is f=0.)
Create a new empty operation table.
Copy the header row of column headers from the old to the new table.
Do for each row of the old table the following:
* Find the column where the entry f is located and take its header value.
* Set the row header of this row in the new table to the held value of the column header of the old table.
* Furthermore, for all entries of the body of the old table that have the same value as this row header set these entries in the new table to the held value of the column header of the old table.
(Because of having different row headers, the rows in the new table are in a different order then the columns.)
Sort the rows of the new table into the same order as the columns with respect to the headers by rearranging the '''complete rows''' including their headers.
As we see, rearranging the rows does not work if and only if the column permutation in the column with the header e has no fixed point. Yet Damm has proved that in a TA/WTA-quasigroup each column permutation has exactly one fixed point. (See Damm's doctoral dissertation page 104, Lemma 7.2) Consequently, rearranging the rows works in all cases of quasigroups used in Damm algorithm.
--[[User:W96|W96]] ([[User talk:W96|talk]]) 19:10, 18 January 2015 (UTC)
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