Index Type

Matrices in Pacioli are indexed by general index sets, not necessarily integers. Indices are first class language members. They are accessible via functions row_domain and column_domain or via literal syntax x@y, with x an index set and y an index key. For example Foo@key13 or Bar@item42.

The type of an element from index set X is Index(X). For example

Foo@key13 :: Index(Foo)
Bar@item42 :: Index(Bar)

A consequence of the matricization of tensors is that a matrix can have any number of row indices and any number of column indices. A row key or column key is a combination of items from possibly multiple index sets. Compound literal indices are constructed with the % symbol. For example Foo@key13%Bar@item42. The index type lists all index sets. In this case:

Index(Foo, Bar)

Special key _ is the only element of the index of zero index sets:

_ :: Index()

It is used to index the empty row and column domains of scalars and vectors.