# Total array ordering

In APL, a **total array ordering**, or **TAO**, is an ordering on all arrays which is used by Grade and Interval Index. Traditionally ordering is defined only for simple arrays of the same shape, so TAO refers to the extension to nested or boxed arrays of arbitrary shape and rank. While J has had such an ordering since 1996 (release 3.01), total array ordering in APL was first seen in Dyalog APL 17.0.

Dyalog's ordering is not a true total order because it does not handle arrays containing simple scalars other than numbers or characters, such as namespaces or objects. Roger Hui has argued that these scalars are not truly arrays, and are not in the scope of a total array ordering.

The name "total array ordering" is taken partly from the mathematical concept of a total order, which must order any two elements, with elements ordering equally only if they are identical. This concept is transferred to APL by specifying that arrays should only order equally if they match.

## External links

TAO Axioms for Dyalog APL

n_le, a dfn implementation of a total array ordering

APL features [edit]
| |
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Built-ins | Primitive function ∙ Primitive operator ∙ Quad name |

Array model | Shape ∙ Rank ∙ Depth ∙ Bound ∙ Index ∙ Axis ∙ Ravel ∙ Ravel order ∙ Element ∙ Scalar ∙ Vector ∙ Matrix ∙ Simple scalar ∙ Simple array ∙ Nested array ∙ Box ∙ Cell ∙ Major cell ∙ Subarray ∙ Empty array ∙ Prototype |

Concepts and paradigms | Leading axis theory ∙ Scalar extension ∙ Conformability ∙ Scalar function ∙ Pervasion ∙ Glyph ∙ Identity element ∙ Total array ordering |

Errors | LIMIT ERROR ∙ RANK ERROR ∙ SYNTAX ERROR |