Some Pexider-type generalizations of the symmetrized multiplicative functional equation on monoids

The symmetrized multiplicative functional equation on a semigroup is $f(xy)+f(yx)=2f(x)f(y)$, and it is known that such a function must be multiplicative if the co-domain is a field of characteristic different from 2. Here we consider some generalizations including the fully Pexiderized equation $f(xy)+g(yx)=h(x)k(y)$ for four unknown functions $f,g,h,k$. This equation has been solved on groups; here we solve it on monoids. Other related functional equations are also treated.