## Functional equations on semigroups.(English)Zbl 0958.39028

Let $$S$$ be a commutative semigroup and $$\sigma:S\to S$$ an endomorphism such that $$\sigma(\sigma(x)) =x$$ for all $$x\in S$$. The author solves each of the 3 functional equations $\begin{gathered} g(x+y)+ g\bigl(x+ \sigma (y)\bigr)= 2g(x)g(y),\;x,y\in S,\\ g(x+y)+ g\bigl(x+ \sigma (y)\bigr) =2g(x),\;x,y\in S,\\ g(x+y)+ g\bigl(x+ \sigma(y) \bigr)=2g (x)+2g(y),\;x,y\in S.\end{gathered}$ They are versions of d’Alembert’s functional equation, Wilson’s functional equation and the quadratic functional equation. The unknown function $$g$$ is defined on $$S$$. Its range is respectively a quadratically closed field of characteristic $$\neq 2$$, a 2-cancellative abelian group and an abelian group uniquely divisible by 2.
These functional equations have been studied earlier by, e.g. the reviewer [see H. Stetkær, Aequationes Math. 54, No. 1-2, 144-172 (1997; Zbl 0899.39007)] in the case of $$S$$ being an abelian group and the range being the complex numbers.
The essential new features of the present paper are (1) that the underlying set $$S$$ is just a commutative semigroup, and (2) that the ranges allowed are more general structures than $$\mathbb{C}$$.

### MSC:

 39B52 Functional equations for functions with more general domains and/or ranges

Zbl 0899.39007
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