Functions on semigroups with vanishing finite Cauchy differences
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- 作者：Lin Li ; Che Tat Ng
- 关键词：Functional equations ; semigroups ; rings ; Cauchy differences ; free abelian groups ; factorization ; quotient groups
- 刊名：Aequationes Mathematicae
- 出版年：2016
- 出版时间：February 2016
- 年：2016
- 卷：90
- 期：1
- 页码：235-247
- 全文大小：497 KB
- 参考文献：1.Baron K., Kannappan P.L.: On the Cauchy difference. Aequ. Math. 46, 112–118 (1993)MathSciNet CrossRef MATH
2.Guo Q., Li L.: Solutions of the third order Cauchy difference equation on groups. Adv. Difference Equ. 2014(203), 14 (2014)MathSciNet
3.Ng C.T.: Kernels of higher order Cauchy differences on free groups. Aequ. Math. 89, 119–147 (2015)MathSciNet CrossRef MATH
4.Ng C.T., Zhao H.: Kernels of the second order Cauchy differences on groups. Aequ. Math. 86, 155–170 (2013)MathSciNet CrossRef MATH
5.Stetkær, H.: Functional equations on groups. World Scientific Publishing Co. Pte. Ltd (2013). ISBN 978-981-4513-12-8 - 作者单位：Lin Li (1)
Che Tat Ng (2)
1. Department of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang, 314001, People’s Republic of China
2. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada - 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Analysis
Combinatorics - 出版者：Birkh盲user Basel
- ISSN：1420-8903
文摘
Let \({(S,\cdot)}\) be a semigroup, (H, +) an abelian group and \({f: S \to H}\). The first and second order Cauchy differences of f are $$\begin{aligned} C^{1}f(x, y)= & {} f(xy) - f(x) - f(y),\\ C^{2}f(x, y, z)= & {} f(xyz) - f(xy) - f(yz) - f(xz) + f(x) + f(y) + f(z).\end{aligned}$$Higher order Cauchy differences C k f are defined recursively. In the case of H = R, a ring where multiplication is distributive over addition, we show that functions \({f: S\to R}\) with vanishing finite Cauchy differences are closed under multiplication. The equation C k f = 0 is considered for cyclic groups, free abelian groups and other selected quotients of the free groups. Keywords Functional equations semigroups rings Cauchy differences free abelian groups factorization quotient groups Mathematics Subject Classification Primary 39B42 39B52 Secondary 39A70 39B72 Dedicated to Professor Roman Ger on his seventieth birthday