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[x] u@v y [x] u@n y Atop Conjunction

Rank -- depends on the rank of v -- WHY IS THIS IMPORTANT?

Forms the composition (u@v) of verbs u and v .

The resulting verb is applied independently to each cell of the argument(s).

  • Operand u is always executed monadically on the result of each application of v
  • Operand v is executed either monadically or dyadically depending whether  u@v has been called monadically or dyadically
    If the right operand of @ is a noun n, it is equivalent to the constant verb (n"_) which ignores its arguments and produces the result n which in turn becomes the argument to u.
   #@> 'Newton';'Einstein'
6 8

   2 3 <@, 4 5
|2 3 4 5|

The result of u@v is a tacit verb equivalent to  [x] (u@:v)"v y

[x] is used to signify that the phrase holds good whether x is present or not.

See: More Information for a visual comparison of At (@:), Atop (@), Compose (&) and Appose (&:).

Common uses

Implement: f(g(x)) -- the mathematical composition of the two functions: f and g.

   mean=: +/ % #
   cat=: ,&1"1
   ]z=: i.2 3       NB. sample noun
0 1 2
3 4 5
   cat z            NB. appends 1 to each row of z
0 1 2 1
3 4 5 1

   mean@cat z       NB. mean of the ROWS of cat z
1 3.25

   mean@cat b.0     NB. rank of (mean@cat)
1 1 1

You can also implement: f(g(x)) with: Compose (&) or Cap ([:) or At (@:).

But see Rank in a hurry: an insidious rank problem for how and when these different methods give different results.

Related Primitives

At (@:), Compose (&), Appose (&:), Hook ((u v)), Fork ((f g h))

More Information

1. Phrase  (u@:v)"v means "apply v followed by u on each cell of the operand(s) independently". The rank of a cell is given by the rank of the verb v . The results from the cells are collected to produce the overall result of  u@v .

Note: unlike At (@:), the rank of Atop (u@v) depends on the ranks of u and v.

2. The difference between At (u@:v) and Atop (u@v) is shown in the first two columns of the diagram below.


The above diagram also shows Appose (&:) and Compose (&) for overall comparison.

3. So what's the difference between Atop (@) and Compose (&) ?

None at all, for the monads (u@v) and (u&v)

  u&v y ↔ u v y
  u@v y ↔ u v y
But the dyads are different
  x u&v y ↔ (v x) u (v y)
  x u@v y ↔ u x v y

According to the J Dictionary -- &: is equivalent to & except that the ranks of the resulting function are infinite; the relation is similar to that between @: and @


The J Dictionary's definition of [x] u@v y as equivalent to  u [x] v y has caused much confusion, because the statement must be read in the context of the rank of v .

The precise definition is as above.