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<p>Motivated by the potential application of formal concept analysis in supporting learning, we attempt to take advantage of the relations between objects to assist in the problem recommendation. This paper is to provide a preliminary preparation for the establishment of a mathematical model and to focus on the theoretical aspects. To achieve this, first, we propose <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10596_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {C}_{i}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">C</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>-connected relations between objects and introduce the concepts of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10596_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {C}_{i}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">C</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>-connected context. Next, we investigate the closedness of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10596_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {C}_{i}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">C</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>-connectedness regarding some mapping images, subcontexts and products. Finally, we explore the relationship between some special complete lattices and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10596_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {C}_{i}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">C</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>-connected contexts. It is proved that the concept lattices generated by <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10596_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {C}_{i}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">C</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>-connected contexts can be represented by some atomistic complete lattices.</p>

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Concept lattices of \(\mathbb {C}_{i}\)-connected contexts and the characterization theorem

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • Zhenhua Jia,
  • Lankun Guo,
  • Mingjie Cai,
  • Qingguo Li

摘要

Motivated by the potential application of formal concept analysis in supporting learning, we attempt to take advantage of the relations between objects to assist in the problem recommendation. This paper is to provide a preliminary preparation for the establishment of a mathematical model and to focus on the theoretical aspects. To achieve this, first, we propose \(\mathbb {C}_{i}\) C i -connected relations between objects and introduce the concepts of \(\mathbb {C}_{i}\) C i -connected context. Next, we investigate the closedness of \(\mathbb {C}_{i}\) C i -connectedness regarding some mapping images, subcontexts and products. Finally, we explore the relationship between some special complete lattices and \(\mathbb {C}_{i}\) C i -connected contexts. It is proved that the concept lattices generated by \(\mathbb {C}_{i}\) C i -connected contexts can be represented by some atomistic complete lattices.