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An involution is a function $f:X\to X$ such that $f\circ f=\text{id}$. Is there a name for a function $g:X\to X$ such that $f\equiv g\circ g$ is an involution? An example is multiplication by $\pm i$ in the complex plane (or more generally for an algebra over $\mathbb{C}$, or for some space which is a product with $\mathbb{C}$). In an almost complex structure, the linear map $J$ with $J^2=-\text{id}$ is also an example. Yet another example is a (properly normalized) Fourier transform, where squaring the Fourier transform $\mathscr{F}$ gives the involution $\mathscr{F}^2[f(t)]=f(-t)$ for square-integrable functions.

In a group, generally, this is clearly a 4-cycle. But considering the connection with complex and almost-complex structures (and quaternions, which have multiplication by $\pm i,\pm j, \pm k$ as 4-cycles) I thought there may be a special name for such a function on a more general space.

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I'm afraid I've only always used "of order (dividing) four" for those. – Hagen von Eitzen Jun 25 '13 at 20:46
Involulution. (random characters for my answer to be submittable) – oxeimon Jun 26 '13 at 6:46

I've never heard a special word for this. Things whose $n$th power is $1$ are usually just called $n$th roots of unity, but perhaps someone employed a special name in some context.

I'm sorely tempted to call it a "spinvolution" because of your two examples. In mathematical physics, there are things (most everything we interact with) that are invariant under a rotation of $2\pi$, and then there are other quantities called spinorial quantities which transform to their negative under a rotation of $2\pi$. Both of your examples really lend themselves to this "spinor" picture :)

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The Hilbert transform $\mathcal{H}$ is sometimes said to be an anti-involution, as $\mathcal{H(H(u))}=-u$ (see Hilbert, inverse transform).

I see this as a sub-case for your question only. I have witnessed $n$-idempotence too, so you could call it $4$-idempotent function.

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I see no reason to do anything substantially different from what rschwieb suggested first in his answer. Perhaps these things have a different name already, but I do not think there is any merit in introducing completely new names for things that are (sort of) well-known.

Let $(A,\cdot)$ be a semigroup and $x,a\in A$. Then $x$ is an $n$-th root of $a$, if $x^n = a$. Note, that $(X^X,\circ)$ is a semigroup (in fact it is a monoid with identity $\rm{id}_X$).

Your $g$ is thus a $4$-th root of $\rm{id}_X$ (with respect to $\circ$, of course). If you want to be more precise, you can call it the $4$-th composition root of $\rm{id}_X$.

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