We consider the functional equation $$G\left( x,G\left( y,x\right) \right) =G\left( y,G\left( x,y\right) \right) $$, posed in Jarczyk and Jarczyk (Aequ Math 72:198–200, 2006). We show that every continuous and reducible solution generates a mean resembling the weighted quasi-arithmetic mean, but no weighted quasi-arithmetic mean is a solution of this equation. This fact particularly implies that the equation is not a direct consequence of the bisymmetry equation and the reflexivity condition. The closedness of the family of solutions with respect to conjugacy is noted. Finally, the translative solutions, homogeneous solutions, and suitable iterative composite functional equation for single variable functions are discussed.

中文翻译：

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在函数方程 $$G\left( x,G\left( y,x\right) \right) = G\left( y,G\left( x,y\right) \right) $$Gx,Gy ,x=Gy,Gx,y 表示

我们考虑函数方程 $$G\left( x,G\left( y,x\right) \right) =G\left( y,G\left( x,y\right) \right) $$, 提出在 Jarczyk 和 Jarczyk (Aequ Math 72:198–200, 2006)。我们表明，每个连续和可约解都会产生一个类似于加权准算术平均值的平均值，但没有加权准算术平均值是这个方程的解。这一事实特别暗示该方程不是双对称方程和自反性条件的直接结果。注意到关于共轭的解决方案族的封闭性。最后讨论了单变量函数的平移解、齐次解和合适的迭代复合函数方程。