Functional equations on semigroups

Summary.

We determine the general solution $g:S\to F$ of the d'Alembert equation¶¶$g(x+y)+g(x+\sigma y)=2g(x)g(y)\qquad (x,y\in S)$,¶the general solution $g:S\to G$ of the Jensen equation¶¶$g(x+y)+g(x+\sigma y)=2g(x)\qquad (x,y\in S)$,¶and the general solution $g:S\to H$ of the quadratic equation¶¶$g(x+y)+g(x+\sigma y)=2g(x)+2g(y)\qquad (x,y\in S)$,¶ where S is a commutative semigroup, F is a quadratically closed commutative field of characteristic different from 2, G is a 2-cancellative abelian group, H is an abelian group uniquely divisible by 2, and $\sigma$ is an endomorphism of S with $\sigma(\sigma(x)) = x$.