Orthogonality Generalized
The concept of orthogonality of Latin squarescan, of course, be generalized.
The following is from A.E. Brouwer, Chapter 5,
"Recursive Constructions of Mutually Orthogonal Latin Squares,"
pp. 149-168 of Latin Squares:
New Developments in the Theory and Applications,
J. Denes and A. D. Keedwell, editors,
Annals of Discrete Mathematics, 46, North-Holland, 1991.
For an earlier (1978) instance
of this generalized
concept of orthogonality, see
Orthogonality of Latin Squares
Viewed as Skewness of Lines
and Latin-Square Geometry.