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 \(\mathbb{C}\).


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


Zbl 0899.39007
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