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What property would you say a function $f(x, y)= z$ has if it is true for all permutations of values $x, y, z$?

Chris_F
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1 Answers1

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Let $f\colon A\to A$ be a function. Assume that for all $x,y,z$ with $f(x,y)=z$, we also have $f(y,x)=z$, $f(x,z)=f(z,x)=y$ and $f(y,z)=f(z,y)=x$.

Equivalently, $f$ is symmetric, i.e., $f(x,y)=f(y,x)$ for all $x,y\in A$, and additionally $f(x,f(x,y))=y$ for all $x,y\in A$, i.e., for every fixed $a\in A$, $f(a,\cdot)$ is involutory.

Hagen von Eitzen
  • 374,180
  • Thanks. Would you happen to know of any examples of two variable functions which are both symmetric and involutory? – Chris_F Jan 16 '21 at 06:19