Consider a set $A$ as well as two binary operations $*_1$ and $*_2$ defined on $A$. Is there a name to describe maps which are defined fully in terms of $*_1$ and $*_2$? For instance $f:A^4\to A$ given by $f(a,b,c,d):=(a*_1 b)*_2(c*_1d).$
In case $A$ is a unitary ring and the two binary operations are $+$ and $\cdot$, the maps I am interested in are essentially polynomial maps whose coefficients are all chosen to be $1$.