We have seen that classic representations of classifications give hierarchical structures, which are the chains of partitions of a partition lattice. But as we have seen also, such structures may change with time, either because of the internal modifications in the description of objects' properties (intrinsic instability), or because of the external increasing or decreasing of the number of objects to be classified (extrinsic instability). For these two reasons, it is clear that it would be pleasant to get an algebra of all the possible transformations of classifications, and particularly explicit rules to explain how we can extend them, without breaking their initial structures.