Regular fractions of factorial arrays

For symmetric arrays of two-level factors, a regular fraction is a well-defined concept, which has been generalized in various ways to arrays of s-level factors with s a prime or prime power, and also to mixed-level arrays with arbitrary numbers of factor levels. This paper introduces three further definitions of a regular fraction for a general array, based on squared canonical correlations or the commuting of projectors. All classical regularity definitions imply regularity under the new definitions, which also permit further arrays to be considered regular. As a particularly natural example, non-cyclic Latin squares, which are not regular under several classical regularity definitions, are regular fractions under the proposed definitions. This and further examples illustrate the different regularity concepts.