Small Latin squares and quasigroups

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.