Weakly associative functions and means - new examples and open questions

In [4], it was observed that each tw-variable weighted quasiarithmetic mean is weakly associative, i.e. it satisfies the equality $$M\left( M\left( x,y\right) ,x\right) =M\left( x,M\left( y,x\right) \right) $$ for all x, y. In the present paper a broader class of non-symmetric weakly associative means is presented. A conjecture that a two-variable formal power series $$M\left( x,y\right) =\sum _{k=1}^{\infty }\sum _{j=0}^{k}a_{k-j,j}x^{k-j}y^{j}$$ with $$a_{1,0}\ne a_{0,1},$$ is weakly associative if and only if $$M\left( x,y\right) =a_{1,0}x+\left( 1-a_{1,0}\right) y$$ is formulated. This conjecture allows to characterize the class of weighted quasiarithmetic means, as well as a new, broader class of means. Looking for translative weakly associative functions we arrive to an open questions concerning the composite functional equation $$\begin{aligned} g\left( g\left( -t\right) +t\right) =g\left( -g\left( t\right) \right) +g\left( t\right) ,\, \ \ \ \ t\in \mathbb {R}. \end{aligned}$$