A set of n − 1 MOLS( n) is called a complete set of MOLS. The MOLS(4) and MOLS(5) examples at the start of this section have been put in standard form.īy putting a set of MOLS( n) in standard form and examining the entries in the second row and first column of each square, it can be seen that no more than n − 1 squares can exist. Using these operations, any set of MOLS can be put into standard form, meaning that the first row of every square is identical and normally put in some natural order, and one square has its first column also in this order. Permuting the entries in any square, independently.Permuting the columns of all the squares simultaneously, and.Permuting the rows of all the squares simultaneously,.The mutual orthogonality property of a set of MOLS is unaffected by The number of mutually orthogonal latin squares the typeface family: serif, sans-serif, slab-serif, cursive, and monospaced.the text: fjords, jawbox, phlegm, qiviut, and zincky.the foreground color: white, red, lime, blue, and yellow.the background color: black, maroon, teal, navy, and silver.Is a representation of the compounded MOLS(5) example above where the four MOLS have the following alphabets, respectively: For example,Īny two of text, foreground color, background color and typeface form a pair of orthogonal Latin squares: In fact, totally different symbol sets can be used for each square of the set of MOLS. In the examples of MOLS given so far, the same alphabet (symbol set) has been used for each square, but this is not necessary as the Graeco-Latin squares show. While it is possible to represent MOLS in a 'compound' matrix form similar to the Graeco-Latin squares, for instance,įor the MOLS(5) example above, it is more typical to compactly represent the MOLS as an orthogonal array (see below). Orthogonal Latin squares were studied in detail by Leonhard Euler, who took the two sets to be S = Orthogonality here means that every pair ( s, t) from the Cartesian product S × T occurs exactly once. A Graeco-Latin square can therefore be decomposed into two orthogonal Latin squares. The arrangement of the s-coordinates by themselves (which may be thought of as Latin characters) and of the t-coordinates (the Greek characters) each forms a Latin square. Graeco-Latin squares (Pairs of orthogonal Latin squares).2.1 The number of mutually orthogonal latin squaresĪ Graeco-Latin square or Euler square or pair of orthogonal Latin squares of order n over two sets S and T (which may be the same), each consisting of n symbols, is an n × n arrangement of cells, each cell containing an ordered pair ( s, t), where s is in S and t is in T, such that every row and every column contains each element of S and each element of T exactly once, and that no two cells contain the same ordered pair.2 Examples of mutually orthogonal Latin squares (MOLS).

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