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Suppose $P$ is some property of some objects and $f$ is a function on those objects. If $Px$ implies $Pf(x)$ and $\lnot Px$ implies $\lnot Pf(x)$, then we might say that "$P$ is invariant under $f$".

Suppose that instead, $P$ is a property such that $Px$ implies $\lnot P f(x)$ and $\lnot Px$ implies $Pf(x)$. What is the correct terminology for this situation?

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  • $\begingroup$ Is there a particular context where you've encountered this property of a property? $\endgroup$ – Ethan Bolker Jun 14 '15 at 16:26
  • $\begingroup$ @EthanBolker $P$ is a property of positions $x$ in a game. For a certain transformation (isomorphism) $f$ on game positions, $f$ reverses property $P$. $\endgroup$ – 6005 Jun 14 '15 at 16:42
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    $\begingroup$ Tempted to suggest "flippant", I instead suggest "anti-invariant" or "reversed", if there isn't a standard term already. $\endgroup$ – Doug Chatham Jun 14 '15 at 17:12
  • $\begingroup$ you might say "P has odd parity under $f$" $\endgroup$ – WW1 Jun 14 '15 at 18:18
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    $\begingroup$ I am all for flippant! $\endgroup$ – GFR Jul 30 '15 at 17:05
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The term anti-invariant seems to be in use for what I believe is this general idea. The Wikipedia glossary of invariant theory defines an anti-invariant as

A relative invariant transforming according to a character of order 2 of a group such as the symmetric group.

which, if my understanding of the relevant terms is correct, is a special case of what I want. Also, an "anti-invariant Riemannian submersion" involves a function not mapping a set to itself, but instead mapping a set to its orthogonal complement (defined e.g. here). Other uses of the term "anti-invariant" can be found by google, e.g. here and here, though it's hard to jump in without context and figure out what the term means.

In conclusion, I think that if one were in great need of a concise term to use repeatedly, anti-invariant would be a good choice. But for most purposes it is probably best to write out "$Px \implies \lnot P f(x)$" explicitly, or in words.

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