Essays/Reflexive

The monad f~ y is defined to be y f y . Such usage also occurs in natural languages, for example self-made billionaire in English or je m'appelle Roger in French. The following are some examples of the reflexive:

`+~`	double
`*~`	square
`/:~`	sort
`?~ n`	random permutation of order `n`
`p}~i.#p`	the inverse of permutation `p`
`i.~`	a more efficient form of self-classify; see Index in Nub
`{./.~`	nub
`#/.~`	frequencies corresponding to the nub; see Histogram
`i.@# = i.~`	nub sieve (the monad `~:`)
`i.~ = i:~`	nub sieve
`#~ i.~ = i:~`	unique items (nub)
`i.&>~@[ i.&\|: i.&>`	index-of for an inverted table
`i.&>~ e.&\|: i.&>~@]`	member-of for an inverted table
`~:@\|:@:(i.&>)~`	nub sieve for an inverted table
`f/~`	function table
`=/~ n$0 1`	`n` by `n` checkerboard
`(f g) y ↔ y (f g) y ↔ (f g)~ y`
`fib=: 3 : 0 " 0` `mp=. +/ .*` `{.{: mp/ mp~^:(I.\|.#:y) 2 2$0 1 1 1x` `)`	the i-th Fibonacci number
`stdarg =: i.@{:@$ , ,:^:(1: -: #@$)` `pvp =: ~. @ (,/) @ ({"1/~)` `subgroup=: pvp^:_ @ stdarg`	subgroup generated by a set of permutations; see Symmetric Array

Contributed by RogerHui.