# What should you call a property, like an invariant, but that is reversed instead of preserved?

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?

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

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.