Close composition

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In SHARP APL and J, a close composition is one of the operators Atop, Over, or Under, defined so that the overall result has function rank equal to the right operand. "Close" is meant in the sense of distance; the left operand is applied nearer to the right operand relative to the corresponding rankless composition, in the sense that it is immediately applied to the result of each individual application of the right operand, as opposed to after the assembly of the right-operand results.[1][2] In SHARP all composition operators follow this pattern, while in J, both close and non-close forms (with a result rank of infinity, matching the definition in most modern APLs) are provided.

Operator SHARP J close J non-close Modern APL
Atop @ @:
Over & &:
Under ¨ &. &.:

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 search
  1. K.E. Iverson. Operators and Functions. §8 Composition and Duality. Research Report RC-7091. IBM Watson Research Center. 1978-04-26.
  2. K.E. Iverson. Rationalized APL. §I. New Operators. Research Report 1. I.P. Sharp Associates. 1983-04-04.