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Insert | m/ u/ _ _ _ | Table

`u/y` applies the dyad`u` between the items of`y` . Thus:
       m=: i. 3 2
       m;(+/m);(+/"1 m);(+/2 3 4)
    +---+---+-----+-+
    |0 1|6 9|1 5 9|9|
    |2 3|   |     | |
    |4 5|   |     | |
    +---+---+-----+-+
`m/y` inserts successive verbs from the gerund`m` between items of `y`,`` extending`m` cyclically as required. Thus,`+`*/i.6` is`0+1*2+3*4+5` . |  | If`x` and`y` are numeric lists, then`x */ y` is their multiplication table. Thus:
       1 2 3 */ 4 5 6 7
     4  5  6  7
     8 10 12 14
    12 15 18 21
In general, each cell of`x` is applied to the entire of`y` .`` Thus`x u/ y` is equivalent to`x u"(lu,_) y` where`lu` is the left rank of`u` . The case`*/` is called outer product in tensor analysis.

If`y` has no items (that is,`0=#y`),`` the result of`u/y` is the neutral or identity element of the function`u`. A neutral of a function`u` is a value`e` such that`x u e` ↔`x` or`e u x` ↔`x`,`` for every`x` in the domain (or some significant sub-domain such as boolean) of`u` .`` This definition of insertion over an argument having zero items extends partitioning identities of the form`u/y` ↔`(u/k{.y) u (u/k}.y)` to the cases`k e. 0,#y` .
The identity function of`u` is a function`ifu` such that`ifu y` ↔`u/y` if`0=#y` .`` The identity functions used are:

Identity function | For
`$&0@}.@$` | `< > + - +. ~: | (2 4 5 6 b.)`
`$&1@}.@$` | `= <: >: * % *. %: ^ ! (1 9 11 13 b.)`
`$&_@}.@$` | `<.`
`$&__@}.@$` | `>.`
`i.@(0&,)@(2&}.)@$` | `,`
`i.@(1&{.)@}.@$` | `C. {`
`=@i.@(1&{.)@}.@$` | `%. +/ . *`
`ifu@#` | `u/`
`$&(v^:_1 ifu$0)@}.@$` | `u&.v`

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