On orthogonally additive mappings. IV
[For part II see the first author, Publ. Math. 35, No.3/4, 241-249 (1988; reviewed above).] Let \(\Phi\) denote a field of characteristic \(\neq 2\), X a \(\Phi\)-vector space of dimension \(\geq 2\) and \((Y,+)\) an abelian group. Furthermore let \(\perp\), which is called orthogonality on X, be a binary relation satisfying certain appropriate conditions on X. The mappings A, Q and F: \(X\to Y\) are said to be additive, quadratic, or orthogonally additive (\(\perp\)-additive), if they satisfy the equations: \(A(x+y)=A(x)+A(y),\) \(x,y\in X\), \(Q(x+y)+Q(x-y)=2Q(x)+2Q(y),\) \(x,y\in X\), or \(F(x+y)=F(x)+F(y),\) \(x,y\in X\), \(x\perp y\), respectively. The main result in this paper is to prove three inclusion relations between the five sets \(\{\) \(A|\) A is additive\(\}\), \(\{\) \(Q|\) Q is quadratic\(\}\), \(\{\) \(F|\) F is \(\perp\)-additive\(\}\), \(\{\) \(D|\) D is odd and \(\perp\)-additive\(\}\) and \(\{\) \(E|\) E is even and \(\perp\)- additive\(\}\). The authors close with some results in the case when the abelian group \((Y,+)\) is 2-torsion-free.
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