Editing Latin square (section)

===Equivalence classes of Latin squares===
{{see also|Small Latin squares and quasigroups}}
Many operations on a Latin square produce another Latin square (for example, turning it upside down).

If we permute the rows, permute the columns, or permute the names of the symbols of a Latin square, we obtain a new Latin square said to be ''[[Quasigroup#Homotopy and isotopy|isotopic]]'' to the first.  Isotopism is an [[equivalence relation]], so the set of all Latin squares is divided into subsets, called ''isotopy classes'', such that two squares in the same class are isotopic and two squares in different classes are not isotopic.

Another type of operation is easiest to explain using the orthogonal array representation of the Latin square.  If we systematically and consistently reorder the three items in each triple (that is, permute the three columns in the array form), another orthogonal array (and, thus, another Latin square) is obtained.  For example, we can replace each triple (''r'',''c'',''s'') by (''c'',''r'',''s'') which corresponds to transposing the square (reflecting about its main diagonal), or we could replace each triple (''r'',''c'',''s'') by (''c'',''s'',''r''), which is a more complicated operation.  Altogether there are 6 possibilities including "do nothing", giving us 6 Latin squares called the conjugates (also [[parastrophe]]s) of the original square.<ref name=DK126>{{harvnb|Dénes|Keedwell|1974|loc=p. 126}}</ref>

Finally, we can combine these two equivalence operations: two Latin squares are said to be ''paratopic'', also ''main class isotopic'', if one of them is isotopic to a conjugate of the other.  This is again an equivalence relation, with the equivalence classes called ''main classes'', ''species'', or ''paratopy classes''.<ref name=DK126 /> Each main class contains up to six isotopy classes.