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I can think of several examples of functions such that twice application of the function is equivalent to no application of it.

  • Additive inverse
  • Multiplicative inverse
  • Fourier transform
  • Complex conjugation
  • Any group built up from $\mathbb{Z}_2$, applying (one of) the $\mathbb{Z}_2$ parts' operation.

"Idempotent" came to mind, but that's wrong. It means $f(f(x)) = f(x)$, not $f(f(x))=x$.

What is the word for this "flip-flop" property?

Grigory M
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isomorphismes
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    Involution – bzc Nov 26 '11 at 19:12
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    also known as '$f$ is its own inverse' – Chris Taylor Nov 26 '11 at 19:13
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    @BrandonCarter Why did you comment instead of answering? I'm not sure whether to "award" the checkmark to you or Leandro. – isomorphismes Nov 26 '11 at 19:41
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    Fourier transform doesn't have that property ;) You get an extra reflection.... – N. S. Nov 26 '11 at 20:21
  • @N.S. Can you explain that? I thought F.T. is its own inverse. – isomorphismes Nov 26 '11 at 21:01
  • @LaoTzu Is not, it is "almost" it's own inverse. $\widehat{\widehat{f}}(x)=f(-x)$.... Keep in mind that for characters $-\chi$ is the same thing as complex conjugation, that's why usually this is written as "the inverse Fourier Transform has an exta conjugate"... – N. S. Nov 26 '11 at 23:17
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    Relatedly, you have involutory matrices, matrices that are the "square roots" of the identity matrix. – J. M. ain't a mathematician Nov 27 '11 at 00:28
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    @LaoTzu I don't feel that I deserve reputation for a one word response; commenting is quicker than making an answer community wiki. – bzc Nov 27 '11 at 06:18
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    possible duplicate of Functions that are their own inversion. – Najib Idrissi Aug 31 '15 at 08:39
  • @NajibIdrissi: You do realize that this question was asked in 2011 and the one that you consider to be the original was asked in 2015, do you? Which one do you think is the original and which one the duplicate, then? – Alex M. Aug 31 '15 at 11:42
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    @AlexM. It's irrelevant. I took in consideration the quality of the questions and most importantly their answers into account. – Najib Idrissi Aug 31 '15 at 11:43
  • @NajibIdrissi: While your argument has its merit, it amounts to redefining the word "duplicate". I suggest that we use it in the precise sense that it has in any dictionary and conserve our originality for more significant tasks. – Alex M. Aug 31 '15 at 11:49
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    @AlexM. You're the one being pedant about definitions. The time two questions were asked is irrelevant when considering which to close as a duplicate of the other. Only the usefulness of both is. Why do you think it's possible to close an older question as a duplicate of a newer one? It's not an oversight. If you're unhappy, open a meta thread. – Najib Idrissi Aug 31 '15 at 11:51

2 Answers2

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Involution is the most common name. They are so fundamental that an entire book has been written on them, the Book of Involutions. I often emphasize their essential role both here and various other places. One should always strive to bring to the fore the innate symmetries in problems, and involutions are one of the simplest examples.

Note: you could have found the answer simply by Googling "self inverse function". The first match is the "self inverse" section of the Wikipedia page on inverse functions, which states "such a function is called an involution".

Community
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Bill Dubuque
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    Self inverse. I didn't think of that. I googled for something approximately like the title. (the word self inverse didn't come readily to mind because I was thinking $f^2=f^0$ rather than $f^1 = f^{-1}$) – isomorphismes Nov 26 '11 at 21:02
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    @Lao It is well-worth the effort to learn how to compose effective searches, by means of which you can truly stand on the shoulder of giants - exposing many beautiful mathematical vistas. – Bill Dubuque Nov 26 '11 at 21:09
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I think this is called an "Involution".

student
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