Major cell

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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 ¯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.


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

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

      ¯1 A
10 15 20 56 3 12 152 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

APL features [edit]
Built-ins Primitive functionPrimitive operatorQuad name
Array model ShapeRankDepthBoundIndex (Indexing) ∙ AxisRavelRavel orderElementScalarVectorMatrixSimple scalarSimple arrayNested arrayCellMajor cellSubarrayEmpty arrayPrototype
Data types Number (Boolean, Complex number) ∙ Character (String) ∙ BoxNamespace
Concepts and paradigms Leading axis theoryScalar extensionConformabilityScalar functionPervasionGlyphIdentity elementComplex floorTotal array ordering