Total array ordering: Difference between revisions

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 array|nested]] or [[box]]ed 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 scalar]]s other than [[number]]s or [[character]]s, such as [[namespace]]s or [[object]]s. [[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 [[wikipedia:total order|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]].

[https://www.jsoftware.com/papers/TAOaxioms.htm TAO Axioms] for [[Dyalog APL]]

[http://dfns.dyalog.com/n_le.htm n_le], a [[dfn]] implementation of a total array ordering

[https://code.jsoftware.com/wiki/Essays/The_TAO_of_J The TAO of J]