<p>In this note, we construct a convolution lattice that is not only a sublattice of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11083_2025_9703_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$(\textbf{m}^{\textbf{n}},\le _{\sqcup })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="bold">m</mi> <mi mathvariant="bold">n</mi> </msup> <mo>,</mo> <msub> <mo>≤</mo> <mo>⊔</mo> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> but also isomorphic to the pentagon. As a result, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11083_2025_9703_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$(\textbf{m}^{\textbf{n}},\le _{\sqcup })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="bold">m</mi> <mi mathvariant="bold">n</mi> </msup> <mo>,</mo> <msub> <mo>≤</mo> <mo>⊔</mo> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> determined by <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11083_2025_9703_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\sqcup $</EquationSource> <EquationSource Format="MATHML"><math> <mo>⊔</mo> </math></EquationSource> </InlineEquation> is not distributive for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11083_2025_9703_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$m,n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, which proves a conjecture related to the distributivity of convolution lattice. Furthermore, we verify that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11083_2025_9703_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$(\textbf{2}^{\textbf{n}},\le _{\sqcup })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mn mathvariant="bold">2</mn> <mi mathvariant="bold">n</mi> </msup> <mo>,</mo> <msub> <mo>≤</mo> <mo>⊔</mo> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is distributive for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11083_2025_9703_Article_IEq6.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11083_2025_9703_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$(\textbf{m}^{\textbf{2}},\le _{\sqcup })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="bold">m</mi> <mn mathvariant="bold">2</mn> </msup> <mo>,</mo> <msub> <mo>≤</mo> <mo>⊔</mo> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is not distributive for <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11083_2025_9703_Article_IEq8.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$m\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Proof of a Conjecture on the Distributivity of Convolution Lattices

期刊论文
发表日期：

2025年7月19日

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Zhi-qiang Liu

摘要

In this note, we construct a convolution lattice that is not only a sublattice of $(\textbf{m}^{\textbf{n}},\le _{\sqcup })$ $(m^{n}, \leq_{⊔})$ but also isomorphic to the pentagon. As a result, $(\textbf{m}^{\textbf{n}},\le _{\sqcup })$ $(m^{n}, \leq_{⊔})$ determined by $\sqcup $ $⊔$ is not distributive for $m,n\ge 3$ $m, n \geq 3$ , which proves a conjecture related to the distributivity of convolution lattice. Furthermore, we verify that $(\textbf{2}^{\textbf{n}},\le _{\sqcup })$ $(2^{n}, \leq_{⊔})$ is distributive for $n\ge 2$ $n \geq 2$ , and $(\textbf{m}^{\textbf{2}},\le _{\sqcup })$ $(m^{2}, \leq_{⊔})$ is not distributive for $m\ge 3$ $m \geq 3$ .