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Encyclopaedia of DesignTheory: Latin squares

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Latin squares as incidence structures

There is no incidence structure whose derived chamber system is a Latin square.

Nevertheless, there is a closely related incidence structure called a net, Associated with a Latin square of order n is a net of order n and degree 3, whose n2 points are the cells of the Latin square and whose 3n blocks are indexed by the rows, columns, and symbols; a point is incident with the row and column containing it and the symbol it contains. For example, given the following Latin square (with cells numbered from 1 to 9),

1 2 3
4 5 6
7 8 9
the point set of the net is {1,2,...,9}, and the blocks are {1,2,3}, {4,5,6}, {7,8,9}, {1,4,7}, {2,5,8}, {3,6,9}, {1,6,8}, {2,4,9}, {3,5,7}.

Conversely, if we are given a net of order n and index 3, the blocks fall into three parallel classes, and we can produce a Latin square as follows: number the blocks in each class from 1 to n, and put symbol k in row i and column j of the square if and only if the unique point of the net lying on the ith block of the first parallel class and the jth block of the second also lies on the kth block of the third.


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Peter J. Cameron
16 April 2002