Solution of Finite Cauchy difference equations on Free Abelian Group
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Abstract
Let (X, .) be a semigroup, (Y, +) an abelian group and g:X→Y. The first and second order Cauchy differences of g are D1g (a,b) = g (ab) − g (a) − g (b), D2g (a,b,c) = g (abc) – g (ab) – g(bc) − g (ac)+ g(a)+ g (b)+ g (c). Finite order Cauchy differences Dnf are defined recursively. In the case of Y = Z, a ring where multiplication is distributive over addition, we show that functions g : X® Z with finite Cauchy differences are closed under multiplication. The equation Dng = 0 is considered for finite abelian groups
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