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<p>Independent systems are a broad class of combinatorial structures. <i>k</i>-extendible systems and <i>k</i>-systems are two important subclasses of independent systems. This paper introduces <i>k</i>-replaceable systems, a novel class of independent systems designed to address limitations in modeling problems that are inadequately captured by <i>k</i>-systems. We thoroughly explore the relationships among <i>k</i>-replaceable systems, <i>k</i>-systems, and <i>k</i>-extendible systems. Specifically, we show that <i>k</i>-extendible systems are strictly contained within <i>k</i>-replaceable systems, while no containment exists between <i>k</i>-systems and <i>k</i>-replaceable systems. Furthermore, by introducing the concepts of minimal replaceable sets and public replaceable sets, we establish conditions under which <i>k</i>-replaceable systems are equivalent to <i>k</i>-extendible systems, as well as conditions where <i>k</i>-replaceable systems can be viewed as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40305_2025_640_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(k\eta \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mi>η</mi> </mrow> </math></EquationSource> </InlineEquation>-systems or <i>k</i>-systems, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40305_2025_640_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">\(1\leqslant \eta \leqslant k\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>⩽</mo> <mi>η</mi> <mo>⩽</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Introducing k-Replaceable Systems: A New Framework in Independent Systems

  • 期刊论文

  • 发表日期：

    2025年9月27日

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  • Qing-Qin Nong,
  • Xin Qin,
  • Su-Ning Gong,
  • Xiao-Ying Qu,
  • Qi-Zhi Fang

摘要

Independent systems are a broad class of combinatorial structures. k-extendible systems and k-systems are two important subclasses of independent systems. This paper introduces k-replaceable systems, a novel class of independent systems designed to address limitations in modeling problems that are inadequately captured by k-systems. We thoroughly explore the relationships among k-replaceable systems, k-systems, and k-extendible systems. Specifically, we show that k-extendible systems are strictly contained within k-replaceable systems, while no containment exists between k-systems and k-replaceable systems. Furthermore, by introducing the concepts of minimal replaceable sets and public replaceable sets, we establish conditions under which k-replaceable systems are equivalent to k-extendible systems, as well as conditions where k-replaceable systems can be viewed as \(k\eta \) k η -systems or k-systems, where \(1\leqslant \eta \leqslant k\) 1 ⩽ η ⩽ k .