Let S be a semigroup, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1226_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma :S \rightarrow S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation> an involutive automorphism or just a surjective homomorphism, and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1226_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">F</mi> </math></EquationSource> </InlineEquation> a field of characteristic <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1226_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$\ne 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≠</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We study solutions <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1226_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="189" /> </InlineMediaObject> <EquationSource Format="TEX">$f,g_2, g_3,h_1,h_2, h_3 : S \rightarrow \mathbb {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>,</mo> <msub> <mi>g</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>g</mi> <mn>3</mn> </msub> <mo>,</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>h</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>h</mi> <mn>3</mn> </msub> <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="Equ52"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1226_Article_Equ52.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="419" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} f(x\sigma (y)) = f(x)h_1(y) + g_2(x)h_2(y) + g_3(x)h_3(y), \ x,y \in S. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mi>σ</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>h</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>g</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>h</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>g</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>h</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="4pt" /> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>S</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>We show that if f is central then it is abelian, and find criteria on S and the equation for f to be central. We apply the results to trigonometric addition and subtraction laws on semigroups.

On centrality of solutions of generalized addition and subtraction laws

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2025年9月17日

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Henrik Stetkær

摘要

Let S be a semigroup, $\sigma :S \rightarrow S$ $σ : S \to S$ an involutive automorphism or just a surjective homomorphism, and $\mathbb {F}$ $F$ a field of characteristic $\ne 2$ $\neq 2$ . We study solutions $f,g_2, g_3,h_1,h_2, h_3 : S \rightarrow \mathbb {F}$ $f, g_{2}, g_{3}, h_{1}, h_{2}, h_{3} : S \to F$ of the functional equation $\begin{aligned} f(x\sigma (y)) = f(x)h_1(y) + g_2(x)h_2(y) + g_3(x)h_3(y), \ x,y \in S. \end{aligned}$ $\begin{matrix} f (x σ (y)) = f (x) h_{1} (y) + g_{2} (x) h_{2} (y) + g_{3} (x) h_{3} (y), x, y \in S . \end{matrix}$ We show that if f is central then it is abelian, and find criteria on S and the equation for f to be central. We apply the results to trigonometric addition and subtraction laws on semigroups.