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Abstract <p> It is proved that, up to natural equivalence, there exist exactly five locally associative analytic functions of two variables: <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(0\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(x\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(y\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((x+y)\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(xy\)</EquationSource> </InlineEquation>. A description of all associative (nonlocally) polynomials is given. All distributive pairs where at least one of the functions is locally associative are also described. </p>

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Associativity and Distributivity of Analytic Functions

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • V. K. Beloshapka

摘要

Abstract

It is proved that, up to natural equivalence, there exist exactly five locally associative analytic functions of two variables: \(0\) , \(x\) , \(y\) , \((x+y)\) , and \(xy\) . A description of all associative (nonlocally) polynomials is given. All distributive pairs where at least one of the functions is locally associative are also described.