2.4.1 Indexed index sets

Index sets occurring in indexing terms can be themselves indexed. (We have already seen an example above in set of sets.) Let us define an index set I and based on it an indexed index set J = {Ji} where Ji is an index set and i I. The elements of J are index sets themselves. In this case the index mechanism is not applied to a singleton but to a set. As an example, suppose I = {a,b}, Ja = {1, 2}, and Jb = {2, 3} then J = {{1, 2},{2, 3}}. Based on these two sets I and J, one can form the following indexed notation:

    (        )
⊙    ⊙                  ⊙
         Ei,j    or          Ei,j
i∈I   j∈Ji             i∈I, j∈Ji
(7)

It is important to note, however, that the index i in i I is an active index, whereas the i in Ji is a passive index. Now (7) can be interpreted as follows:

 ⊙
      Ei,j  with  K  ⊆ I × (J1 ∪ ⋅⋅⋅ ∪ J|I|)
(i,j)∈K
(8)

This means that the concept of indexed index set can be reduced to compound index set and does not add any new features. Nevertheless, a notation like (7) may sometimes be more convenient, because the tuples are presented as a hierarchical structure instead of a flat list.