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In a similar fashion to the partitioning of the integers into residue classes, integer spaces (finitely generated free Z-modules) can be partitioned into complete or incomplete vector congruence classes. Complete vector congruence classes are computable by an application of the Smith normal form. The problem of computing efficiently incomplete vector congruence classes is discussed and a few methods are suggested. The application of complete and incomplete vector congruence classes is in performing set-theoretic operations on integer spaces. All set-theoretic operations can be performed on integer spaces which partition into complete vector congruence classes. Some set-theoretic operations can also be performed on integer spaces which partition into incomplete vector congruence classes, but not always in a satisfactory manner.

The computation and application of vector congruence classes