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.
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.
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.
[2] 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.
[5] 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.
To stress this point, small Latin squares sometimes use letters of the alphabet as a symbol set.
Thus, as Latin squares, these should be considered the same: Similarly, and for the same reason, should be thought of as the same.
[9] One can normalize a Cayley table of a quasigroup in the same manner as a reduced Latin square.
Some of these counts are the same for every isotope of a Latin square and are referred to as isotopy invariants.
Another is the total number of transversals, a set of n positions in a Latin square of order n, one in each row and one in each column, that contain no element twice.
Latin squares with different values for these counts must lie in different isotopy classes.
The reduced square is in an isomorphism class with three elements and so the corresponding quasigroup is a loop, in fact it is a group,
Of the 576 Latin squares, 288 are solutions of the 4×4 version of Sudoku, sometimes called Shi Doku [1].
[10] The first square has 15 transversals, no intercalates and is the unbordered Cayley table of the cyclic group
It represents a loop that is not a group, since, for instance, 2 + (3 + 4) = 2 + 0 = 2, while (2 + 3) + 4 = 0 + 4 = 4, so the associative law does not hold.
The counting of Latin squares has a long history, but the published accounts contain many errors.
Euler in 1782,[11] and Cayley in 1890,[12] both knew the number of reduced Latin squares up to order five.
In 1915, MacMahon[13] approached the problem in a different way, but initially obtained the wrong value for order five.
M. Frolov gave an incorrect count of reduced squares of order seven.
R.A. Fisher and F. Yates,[17] unaware of earlier work of E. Schönhardt,[18] gave the number of isotopy classes of orders up to six.
In 1939, H. W. Norton found 562 isotopy classes of order seven,[19] but acknowledged that his method was incomplete.