Weakly associative functions

Let $$I\subset \mathbb {R}$$ be an interval. A function $$M:I^{2}\rightarrow I$$ is said to be weakly associative, if $$\begin{aligned} M\left( M\left( x,y\right) ,x\right) =M\left( x,M\left( y,x\right) \right) , \qquad x,y\in I. \end{aligned}$$ One can easily check that every weighted quasi-arithmetic mean, i.e. a function $$M:I^{2}\rightarrow I$$ given by $$ M\left( x,y\right) =f^{-1}\left( pf\left( x\right) +\left( 1-p\right) f\left( y\right) \right) , $$ where $$f:I\rightarrow \mathbb {R}$$ is a continuous and strictly monotonic function and $$p\in \left[ 0,1\right] $$ , satisfies the above condition, so it is weakly associative. We give the characterization of weakly associative functions in the class of some generalized weighted quasi-arithmetic means. Moreover, we characterize premeans which are rational functions of degree at most 2 and weakly associative.