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
>:@#.~/.~&.q:	sum of divisors
(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 Roger Hui.