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Given a binary relation $r$, its reflexive closure, $r \cup id$, is sometimes written as $r^?$ or $r^=$. Its transitive closure is written as $r^+$. Its reflexive, transitive closure is written as $r^*$.

What about its symmetric closure, $r \cup r^{-1}$? Is there any existing notation for that? I saw $s(r)$ once, but that's not particularly appealing to me. Are there any other candidates?

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  • $\begingroup$ I've never seen one. I don't visit places where I would be likely to, so haven't seen the others either. If you need the concept often in something you are writing, invent notation and tell your reader. $\endgroup$
    – Ethan Bolker
    May 29, 2019 at 12:38

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I would use $R^s$ for the symmetrisation of a relation $R$, as in this link.

By symmetrization of the binary relation $R$ on a set one obtains the symmetric relation $R^s$ defined by: $a \mathrel{R^s} b$ if and only if $a \mathrel{R} b$ or $b \mathrel{R} a$, such that every symmetric relation implying $\mathrel{R}$ also implies $\mathrel{R^s}$.

Zassenhaus, Hans J. The theory of groups. 2nd ed. Chelsea Publishing Company, New York, 1958. 265 pp.

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  • $\begingroup$ Thanks! I’m afraid the link doesn’t work for me though. I presume it shows the notation in use in a book. Perhaps you could kindly attach a screenshot instead? Thanks again. $\endgroup$
    – John Wickerson
    May 29, 2019 at 19:17
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    $\begingroup$ I have reproduced the relevant paragraph. $\endgroup$
    – J.-E. Pin
    May 30, 2019 at 8:12
  • $\begingroup$ Very kind, thank you $\endgroup$
    – John Wickerson
    May 30, 2019 at 12:16

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