On the functional equation x + f(y + f(x))=y + f(x + f(y)) |
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Authors: | Jürg Rätz |
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Institution: | 1. Mathematisches Institut der Universit?t Bern, Sidlerstrasse 5, 3012, Bern, Switzerland
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Abstract: | For an abelian group (G, + ,0) we consider the functional equation $$f : G \to G, x + f(y + f(x)) = y + f(x + f(y)) \quad (\forall x, y \in G), \quad\quad\qquad (1)$$ most times together with the condition $$f(0) = 0.\qquad\qquad\qquad\qquad\qquad (0)$$ Our main question is whether a solution of ${(1) \wedge (0)}$ must be additive, i.e., an endomorphism of G. We shall answer this question in the negative (Example 3.14) Rätz (Aequationes Math 81:300, 2011). |
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