Latin squares and finite quasigroups are equivalent mathematical objects, although the former has a combinatorial nature while the latter is more algebraic. The listing below will consider the examples of some very small orders, which is the side length of the square, or the number of elements in the equivalent quasigroup.
The equivalence Given a quasigroup Q with n elements, its Cayley table (almost universally called its multiplication table) is an (n + 1) × (n + 1) table that includes borders; a top row of column headers and a left column of row headers. Removing the borders leaves an n × n array that is a Latin square. This process can be reversed, starting with a Latin square, introduce a bordering row and column to obtain the multiplication table of a quasigroup. While there is complete arbitrariness in how this bordering is done, the quasigroups obtained by different choices are sometimes equivalent in the sense given below.
Isotopy and isomorphism Two Latin squares, L1 and L2 of order n (that is, they are n × n {\displaystyle n\times n} squares) are isotopic if there are three bijections from the rows, columns and symbols of L1 onto the rows, columns and symbols of L2, respectively, that map L1 to L2. Isotopy is an equivalence relation and the equivalence classes are called isotopy classes. A stronger form of equivalence exists. Two Latin squares L1 and L2 of side n with common symbol set S that is also the index set for the rows and columns of each square are isomorphic if there is a bijection g: S → S such that g(L1(i, j)) = L2(g(i), g(j)) for all i, j in S. An alternate way to define isomorphic Latin squares is to say that a pair of isotopic Latin squares are isomorphic if the three bijections used to show that they are isotopic are, in fact, equal. Isomorphism is also an equivalence relation and its equivalence classes are called isomorphism classes. An alternate representation of a Latin square is given by an orthogonal array. For a Latin square of order n this is an n2 × 3 matrix with columns labeled r, c and s and whose rows correspond to a single position of the Latin square, namely, the row of the position, the column of the position and the symbol in the position. Thus for the order three Latin square,
the orthogonal array is given by:
The condition for an appropriately sized matrix to represent a Latin square is that for any two columns the n2 ordered pairs determined by the rows in those columns are all the pairs (i, j) with 1 ≤ i, j ≤ n, once each. This property is not lost by permuting the three columns (but not the labels), so another orthogonal array (and thus, another Latin square) is obtained. This is the operation of choosing a single consistent way to permute each triplet (r, c, s) where r is the row index, c the column index, and s the symbol number. For example, by permuting the first two columns, which corresponds to transposing the square (reflecting about its main diagonal) gives another Latin square, which may or may not be isotopic to the original. In this case, if the quasigroup corresponding to this Latin square satisfies the commutative law, the new Latin square is the same as the original one. Altogether there are six possibilities including "do nothing", giving at most six Latin squares called the conjugates (also parastrophes) of the original square. 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 also an equivalence relation, with the equivalence classes called main classes, species, or paratopy classes. Each main class contains up to six isotopy classes. A main class is a disjoint union of isotopy classes and an isotopy class is a disjoint union of isomorphism classes.
Isotopic quasigroups Let (Q,∘) and (R,∗) be two quasigroups. An ordered triple (f, g, h) of bijections from Q onto R is called an isotopism of (Q,∘) onto (R,∗) if f(x) ∗ g(y) = h(x ∘ y) for all x, y in G. Such quasigroups are said to be isotopic. If in the above definition f = g = h then the quasigroups are said to be isomorphic. Unlike the situation with Latin squares, when two isotopic quasigroups are represented by Cayley tables (bordered Latin squares), the permutations f and g operate only on the border headings and do not move columns and rows, while h operates on the body of the table. Permuting the rows and columns of a Cayley table (including the headings) does not change the quasigroup it defines, however, the Latin square associated with this table will be permuted to an isotopic Latin square. Thus, normalizing a Cayley table (putting the border headings in some fixed predetermined order by permuting rows and columns including the headings) preserves the isotopy class of the associated Latin square. Furthermore, if two normalized Cayley tables represent isomorphic quasigroups then their associated Latin squares are also isomorphic. Hence, the number of distinct quasigroups of a given order is the number of isomorphism classes of Latin squares of that order.
Notation The set of symbols used in a Latin square (or quasigroup) is arbitrary and individual symbols carry no meaning, even if they may have a meaning in other contexts. Thus, since it is most common to see the symbol sets {1, 2, ..., n} or {0, 1, ..., n − 1} used, one must remember that these symbols carry no numerical meaning. To stress this point, small Latin squares sometimes use letters of the alphabet as a symbol set.
Counting Latin squares As a Latin square is a combinatorial object, the symbol set used to write the square is immaterial. Thus, as Latin squares, these should be considered the same:
a b b a {\displaystyle {\begin{matrix}a&b\\b&a\end{matrix}}} and 1 2 2 1 . {\displaystyle {\begin{matrix}1&2\\2&1\end{matrix}}.}
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