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orthogonal Latin squares
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(Definition)
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Given two Latin squares
 and
 of the same order  , we can combine them coordinate-wise to form a single square, whose cells are ordered pairs of elements from  and  respectively. Formally, we can form a
function
 given by
This function  says that we have created a new square  , whose cell  contains the ordered pair of values, the first coordinate of which corresponds to the value in cell  of  , and the second to the value in cell  of  . We may write the combined square  .
For example,
In general, the combined square is not a Latin square unless the original two squares are equivalent:
 iff
 . Nevertheless, the more interesting aspect of pairing up two Latin squares (of the same order) lies in the function  :
Definition. We say that two Latin squares are orthogonal if  is a bijection.
Since there are  cells in the combined square, and
 , the function  is a bijection if it is either one-to-one or onto. It is therefore easy to see that the two Latin squares in the example above are orthogonal.
Remarks.
- The combined square is usually known as a Graeco-Latin square, originated from statisticians Fischer and Yates.
- It can be shown that if
 are Latin squares of order  such that each pair of them are orthogonal, then  . If the equality occurs, then the set of Latin squares are said to be complete.
- (Bose) If
 , then
 form a complete set of pairwise orthogonal Latin squares of order  iff there exists a finite projective plane of order  .
- 1
- H. J. Ryser, Combinatorial Mathematics, The Carus Mathematical Monographs, 1963
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"orthogonal Latin squares" is owned by CWoo.
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(view preamble)
| Other names: |
mutually orthogonal Latin squares, MOLS, pairwise orthogonal Latin squares |
| Also defines: |
complete set of Latin squares |
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Cross-references: finite projective plane, equality, Graeco-Latin square, easy to see, onto, one-to-one, bijection, pairing, iff, equivalent, coordinate, contains, function, ordered pairs, cells, square, order, Latin squares
This is version 9 of orthogonal Latin squares, born on 2006-07-12, modified 2006-10-01.
Object id is 8138, canonical name is OrthogonalLatinSquares.
Accessed 1921 times total.
Classification:
| AMS MSC: | 05B15 (Combinatorics :: Designs and configurations :: Orthogonal arrays, Latin squares, Room squares) | | | 62K10 (Statistics :: Design of experiments :: Block designs) |
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Pending Errata and Addenda
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