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In the APL [[array model]] and [[leading axis theory]], a '''major cell''', or '''item''', is a [[cell]] of an array which has [[rank]] one smaller than the rank of the array, or equal to it if the array is a [[scalar]]. The number of major cells in an array is its [[Tally]], and a function can be called on the major cells of an array individually by applying it with rank <syntaxhighlight lang=apl inline>¯1</source> using the [[Rank operator]]. Functions designed to follow leading axis theory often manipulate the major cells of an array. For example, [[Reverse First]] (<syntaxhighlight lang=apl inline>⊖</source>) is considered the primary form of [[Reverse]] in leading-axis languages because it can be interpreted as reversing the major cells of its argument; [[J]] removes last-axis Reverse entirely.
In the APL [[array model]] and [[leading axis theory]], a '''major cell''', or '''item''', is a [[cell]] of an array which has [[rank]] one smaller than the rank of the array, or equal to it if the array is a [[scalar]]. The number of major cells in an array is its [[Tally]], and a function can be called on the major cells of an array individually by applying it with rank <syntaxhighlight lang=apl inline>¯1</syntaxhighlight> using the [[Rank operator]]. Functions designed to follow leading axis theory often manipulate the major cells of an array. For example, [[Reverse First]] (<syntaxhighlight lang=apl inline>⊖</syntaxhighlight>) is considered the primary form of [[Reverse]] in leading-axis languages because it can be interpreted as reversing the major cells of its argument; [[J]] removes last-axis Reverse entirely.


== Examples ==
== Examples ==


<syntaxhighlight lang=apl inline>A</source> is an array with [[shape]] <syntaxhighlight lang=apl inline>3 4</source>. Using [[Tally]] we see that the number of major cells in <syntaxhighlight lang=apl inline>A</source> is the first element of the shape, <syntaxhighlight lang=apl inline>3</source>:
<syntaxhighlight lang=apl inline>A</syntaxhighlight> is an array with [[shape]] <syntaxhighlight lang=apl inline>3 4</syntaxhighlight>. Using [[Tally]] we see that the number of major cells in <syntaxhighlight lang=apl inline>A</syntaxhighlight> is the first element of the shape, <syntaxhighlight lang=apl inline>3</syntaxhighlight>:
<syntaxhighlight lang=apl>
<syntaxhighlight lang=apl>
       ⎕←A ← 5 3 1 ∘.∧ 2 3 4 5
       ⎕←A ← 5 3 1 ∘.∧ 2 3 4 5
Line 11: Line 11:
       ≢A
       ≢A
3
3
</source>
</syntaxhighlight>
We can separate <syntaxhighlight lang=apl inline>A</source>'s major cells using [[Enclose]] with [[Rank operator|rank]] <syntaxhighlight lang=apl inline>¯1</source>:
We can separate <syntaxhighlight lang=apl inline>A</syntaxhighlight>'s major cells using [[Enclose]] with [[Rank operator|rank]] <syntaxhighlight lang=apl inline>¯1</syntaxhighlight>:
<syntaxhighlight lang=apl>
<syntaxhighlight lang=apl>
       ⊂⍤¯1 ⊢A
       ⊂⍤¯1 ⊢A
Line 18: Line 18:
│10 15 20 5│6 3 12 15│2 3 4 5│
│10 15 20 5│6 3 12 15│2 3 4 5│
└──────────┴─────────┴───────┘
└──────────┴─────────┴───────┘
</source>
</syntaxhighlight>
Given another array <syntaxhighlight lang=apl inline>B</source> we can search for cells of <syntaxhighlight lang=apl inline>B</source> which [[match]] major cells of <syntaxhighlight lang=apl inline>B</source>. [[High-rank set functions|High-rank]] [[Index-of]] always searches for right argument cells whose rank matches the rank of a left argument major cell: if the right argument is a [[vector]] and not a [[matrix]] then it searches for the entire vector rather than its major cells (which are [[scalar]]s).
Given another array <syntaxhighlight lang=apl inline>B</syntaxhighlight> we can search for cells of <syntaxhighlight lang=apl inline>B</syntaxhighlight> which [[match]] major cells of <syntaxhighlight lang=apl inline>B</syntaxhighlight>. [[High-rank set functions|High-rank]] [[Index-of]] always searches for right argument cells whose rank matches the rank of a left argument major cell: if the right argument is a [[vector]] and not a [[matrix]] then it searches for the entire vector rather than its major cells (which are [[scalar]]s).
<syntaxhighlight lang=apl>
<syntaxhighlight lang=apl>
       ⎕←B ← ↑ 4,/⍳6
       ⎕←B ← ↑ 4,/⍳6
Line 29: Line 29:
       A ⍳ 2 3 4 5
       A ⍳ 2 3 4 5
3
3
</source>
</syntaxhighlight>


{{APL features}}[[Category:Array relationships]]
{{APL features}}[[Category:Array relationships]]

Revision as of 21:18, 10 September 2022

In the APL array model and leading axis theory, a major cell, or item, is a cell of an array which has rank one smaller than the rank of the array, or equal to it if the array is a scalar. The number of major cells in an array is its Tally, and a function can be called on the major cells of an array individually by applying it with rank ¯1 using the Rank operator. Functions designed to follow leading axis theory often manipulate the major cells of an array. For example, Reverse First () is considered the primary form of Reverse in leading-axis languages because it can be interpreted as reversing the major cells of its argument; J removes last-axis Reverse entirely.

Examples

A is an array with shape 3 4. Using Tally we see that the number of major cells in A is the first element of the shape, 3:

      ⎕←A ← 5 3 1 ∘.∧ 2 3 4 5
10 15 20  5
 6  3 12 15
 2  3  4  5
      ≢A
3

We can separate A's major cells using Enclose with rank ¯1:

      ⊂⍤¯1 ⊢A
┌──────────┬─────────┬───────┐
│10 15 20 5│6 3 12 15│2 3 4 5│
└──────────┴─────────┴───────┘

Given another array B we can search for cells of B which match major cells of B. High-rank Index-of always searches for right argument cells whose rank matches the rank of a left argument major cell: if the right argument is a vector and not a matrix then it searches for the entire vector rather than its major cells (which are scalars).

      ⎕←B ← ↑ 4,/⍳6
1 2 3 4
2 3 4 5
3 4 5 6
      A ⍳ B
4 3 4
      A ⍳ 2 3 4 5
3


APL features [edit]
Built-ins Primitives (functions, operators) ∙ Quad name
Array model ShapeRankDepthBoundIndex (Indexing) ∙ AxisRavelRavel orderElementScalarVectorMatrixSimple scalarSimple arrayNested arrayCellMajor cellSubarrayEmpty arrayPrototype
Data types Number (Boolean, Complex number) ∙ Character (String) ∙ BoxNamespaceFunction array
Concepts and paradigms Conformability (Scalar extension, Leading axis agreement) ∙ Scalar function (Pervasion) ∙ Identity elementComplex floorArray ordering (Total) ∙ Tacit programming (Function composition, Close composition) ∙ GlyphLeading axis theoryMajor cell searchFirst-class function
Errors LIMIT ERRORRANK ERRORSYNTAX ERRORDOMAIN ERRORLENGTH ERRORINDEX ERRORVALUE ERROREVOLUTION ERROR