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<p>Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((S,+)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a commutative semigroup, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> an involution of <i>S</i>, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathbb {F}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">F</mi> </math></EquationSource> </InlineEquation> a field of characteristic different from 2. We find all solutions <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(f,g: S\rightarrow \mathbb {F}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">F</mi> </mrow> </math></EquationSource> </InlineEquation> of the functional equation <Equation ID="Equ17"> <EquationSource Format="TEX">\( f(x+y)+g(x+\sigma (y))=f(x)+f(y)+g(x)g(y), \quad x,y \in S, \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>S</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation>which arises from summing, side by side, the additive Cauchy equation <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(f(x+y)=f(x)+f(y)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the equation <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(g(x+\sigma (y))=g(x)g(y)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This enables us to investigate whether this equation is equivalent to the system <Equation ID="Equ18"> <EquationSource Format="TEX">\( \left\{ \begin{array}{l} f(x+y)=f(x)+f(y) \\ g(x+\sigma (y))=g(x) g(y) \end{array}\right. \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>for all <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(x,y \in S\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation> (the alienation phenomenon). Additionally, we consider a related problem that couples d’Alembert’s functional equation with the additive Cauchy equation.</p>

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Alienation of additivity and \(\sigma \)-multiplicativity on semigroups

  • 期刊论文

  • 发表日期：

    2026年6月27日

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  • Sara Juiher,
  • Brahim Fadli

摘要

Let \((S,+)\) ( S , + ) be a commutative semigroup, \(\sigma \) σ an involution of S, and \(\mathbb {F}\) F a field of characteristic different from 2. We find all solutions \(f,g: S\rightarrow \mathbb {F}\) f , g : S → F of the functional equation \( f(x+y)+g(x+\sigma (y))=f(x)+f(y)+g(x)g(y), \quad x,y \in S, \) f ( x + y ) + g ( x + σ ( y ) ) = f ( x ) + f ( y ) + g ( x ) g ( y ) , x , y ∈ S , which arises from summing, side by side, the additive Cauchy equation \(f(x+y)=f(x)+f(y)\) f ( x + y ) = f ( x ) + f ( y ) and the equation \(g(x+\sigma (y))=g(x)g(y)\) g ( x + σ ( y ) ) = g ( x ) g ( y ) . This enables us to investigate whether this equation is equivalent to the system \( \left\{ \begin{array}{l} f(x+y)=f(x)+f(y) \\ g(x+\sigma (y))=g(x) g(y) \end{array}\right. \) f ( x + y ) = f ( x ) + f ( y ) g ( x + σ ( y ) ) = g ( x ) g ( y ) for all \(x,y \in S\) x , y ∈ S (the alienation phenomenon). Additionally, we consider a related problem that couples d’Alembert’s functional equation with the additive Cauchy equation.