<p>In this paper, the constructs of a grey <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> minimal and a grey <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> maximal descriptions are proposed, and four categories of grey <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> neighborhoods are established. Later, four kinds of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> neighborhoods extracted from grey <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> neighborhoods are introduced. Subsequently, four models of grey <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> coverings are generated by employing these grey neighborhoods and their interconnections and disparities among the previous versions are examined. In addition, attribute reduction utilizing these mentioned models is explored and the concepts of merge and join reduction of grey <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> covering are proposed. Therefore, we define the merge- and join-reducible object/element. Finally, the topological properties of these grey <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> neighborhoods are studied and a new approach, grey topological space, is proposed.</p>

Some models of grey $\beta $ coverings and their topological properties

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发表日期：

2025年1月27日

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Mohammed Atef,
Sifeng Liu,
Saad Ahmed Javed

摘要

In this paper, the constructs of a grey $\beta $ $β$ minimal and a grey $\beta $ $β$ maximal descriptions are proposed, and four categories of grey $\beta $ $β$ neighborhoods are established. Later, four kinds of $\beta $ $β$ neighborhoods extracted from grey $\beta $ $β$ neighborhoods are introduced. Subsequently, four models of grey $\beta $ $β$ coverings are generated by employing these grey neighborhoods and their interconnections and disparities among the previous versions are examined. In addition, attribute reduction utilizing these mentioned models is explored and the concepts of merge and join reduction of grey $\beta $ $β$ covering are proposed. Therefore, we define the merge- and join-reducible object/element. Finally, the topological properties of these grey $\beta $ $β$ neighborhoods are studied and a new approach, grey topological space, is proposed.

Some models of grey \(\beta \) coverings and their topological properties

摘要