Functional Equations with an Anti-Endomorphism for Functions with Multidimensional Codomains

Let S be a semigroup, ℍ be the skew field of quaternions, and ψ: S → S be an anti-endomorphism. We determine the general solution of the functional equation g(xy)-g(xψ(y))=2g(x)g(y), x, y∈S, g\left( {xy} \right) - g\left( {x\psi \left( y \right)} \right) = 2g\left( x \right)g\left( y \right),\,\,\,\,\,\,\,\,\,\,x,\,y \in S, where g : S →ℂ is the unknown function. And when S = M is a monoid, we solve the functional equation g(xy)+g(xψ(y))=2g(x)g(y), x, y∈M, g\left( {xy} \right) + g\left( {x\psi \left( y \right)} \right) = 2g\left( x \right)g\left( y \right),\,\,\,\,\,\,\,\,x,\,y \in M, where g : M → ℍ is the unknown function.