Reordering transformations
We would need to deal with different orders within sets of objects. Given two different total orders O1 and O2 on the same set A, there is a reordering transformation r (O1, O2) which is a one-to-one correspondence between the set and its permutation. There are several ways to encode reordering which are beyond the scope of this presentation.
We would often need to enumerate cells in a grid. Any space filling curve may be used for this purpose. One frequently used way is the “odometer enumeration”, i.e. sorting by the order of members in the order of dimensions. This is close to the GROUP BY clause in SQL.