Converse to a proposition regarding associative and switchable binary operations.

Question

I define the switchability property of binary operations as follows: An ordered pair $(+,*)$ of binary operations on a set $S$ is said to satisfy switchability if for all $x,y,z$ in $S$, $(x+y)*z=x+(y*z)$. I read in a comment on a previous question that, given an associative operation $+$ on a set $S$, and given an arbitrary unary function $f$ on $S$, if we define a new operation $*$ by stipulating that $x*y=x+f(y)$, then the pair $(+,*)$ satisfies switchability. I am interested in a converse to the previous. My question now is, given an associative operation $+$ on a set $S$, if we are given that an operation $*$ on $S$ such that $(+,*)$ satisfies switchability, must there exist a unary function $f$ on $S$ such that $x*y=x+f(y)$?

Answer 1

This is no complete answer. But under some additional assumptions, which for example are satisfied if $S$ is a group, the converse is true.

Assume that there are some $a, a'\in S$ such that $x+a'+a=x$ for all $x$. Then $x*y=((x+a')+a)*y=x+a' +a*y$.

So $f$ defined as $f(y):=a'+a*y$ has the desired property.