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I'm interested in symmetric functions of two variables $f(x,y)$ with the property that $f(f(x,y),z)$ is symmetric in $x,y,$ and $z$ (or equivalently, symmetric functions such that $f(f(x,y),z) = f(f(x,z),y)$).

a) Given a non-symmetric function $g(x,y)$, we can make a symmetric one $f(x,y) = 0.5*[g(x,y) + g(y,x)]$. Is there an analogous procedure to take a symmetric function $f(x,y)$ and make a new one which has the above property?

b) Is there a better way to characterize functions $f$ which have this property?

Aiden Chow
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Jonathan Shaw
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  • The easiest way is for $f$ to be associative and commutative. Are there examples you're interested in that don't satisfy that property? Note that the symmetry condition is linear, which is why averaging works out. The condition you write down is very nonlinear as a function of $f$ so no analogous averaging argument is possible. – Qiaochu Yuan Sep 02 '20 at 22:56
  • @Qiaochu Good point on associative. It's funny how in thinking of it as a symmetric function instead of a commutative operator, I missed the obvious link with associativity. But regarding a), you're correct that averaging doesn't work, but is there something that does? And b), https://arxiv.org/pdf/1309.7303.pdf makes a reasonable effort (searching on associativity helped!). I'm not sure yet how far it gets me. What I'm ultimately trying to do is train a neural net which is restricted to learning only commutative and associative operators. – Jonathan Shaw Sep 03 '20 at 15:58

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