Scholarlink

<p>Rough set theory has been extensively studied in regard to its lattice structure. However, this article concerns with the (commutative) ring structure of rough set theory. We show that a finite approximation space can be identified by a cube free natural number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10476_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation> by providing an isomorphism between lattice of rough sets and lattice of ideals of the ring <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10476_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {Z}_{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. We introduce a ring structure on the rough sets via the ring structure on ideals of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10476_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {Z}_{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. Moreover, we also classify all the rings which are isomorphic to the rings formed by the rough sets.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Ring structure of rough sets

  • 期刊论文

  • 发表日期：

    2025年1月1日

查看全文
  • Arun Kumar,
  • Bisham Dewan

摘要

Rough set theory has been extensively studied in regard to its lattice structure. However, this article concerns with the (commutative) ring structure of rough set theory. We show that a finite approximation space can be identified by a cube free natural number \(n\) n by providing an isomorphism between lattice of rough sets and lattice of ideals of the ring \(\mathbb {Z}_{n}\) Z n . We introduce a ring structure on the rough sets via the ring structure on ideals of \(\mathbb {Z}_{n}\) Z n . Moreover, we also classify all the rings which are isomorphic to the rings formed by the rough sets.