Abstract We make gradual generalizations in this paper, from the concepts of twin approximation operators in covering rough set theory to the concepts of couple fuzzy covering rough set models in fuzzy rough set theory, and further to the concepts of couple L-fuzzy covering rough set models in L-fuzzy rough set theory. Given a fuzzy covering approximation space ( U , C ˜ ) and β ∈ ( 0 , 1 ] , for each x ∈ U , we divide C ˜ into two parts ⊤ x = { C ˜ i ∈ C ˜ : C ˜ i ( x ) ≥ β } and ⊥ x = { C ˜ i ∈ C ˜ : C ˜ i ( x ) β } . To fully describe x from positive aspect and negative aspect, both the two parts ⊤ x and ⊥ x are important, especially the combination of them. So, in this paper, based on the two parts, we define 1 st couple β-fuzzy covering rough set models [ ( P − ˜ , P + ˜ ) , ( Q − ˜ , Q + ˜ ) ] and 2 nd couple fuzzy β-covering rough set models [ ( P − ‾ , P + ‾ ) , ( Q − ‾ , Q + ‾ ) ] . Both the two types of couple fuzzy β-covering rough set models are generalizations of the twin approximations defined in covering rough set theory. Since each pair of operators in these models are not only closely related but complementary, they can be used to analyze and solve practical problems from positive and negative aspects so as to make a crucial decision. So we then give some examples to show their practical value. The relationships between our models and some other models introduced in previous literature are investigated, and the matrix methods are given to calculate the related approximations and to describe the relationships between every couple models. To generalize the couple fuzzy β-covering rough set models to the CCD lattice are of a bit complicated, because any two elements in the lattice cannot be compared with each other generally. After some effort we successfully construct the ideal couple L-fuzzy β-covering rough set models in L-fuzzy rough set theory which are just the generalizations of the two types of couple fuzzy β-covering rough set models. We also obtain the lattice matrix representations to calculate the related approximations.

中文翻译：

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耦合模糊覆盖粗糙集模型及其对 CCD 点阵的推广

摘要 本文从覆盖粗糙集理论中孪生逼近算子的概念到模糊粗糙集理论中耦合模糊覆盖粗糙集模型的概念，再到耦合L-模糊覆盖粗糙集的概念，逐步推广。 L-模糊粗糙集理论中的模型。给定一个模糊覆盖逼近空间 ( U , C ˜ ) 和 β ∈ ( 0 , 1 ] ，对于每个 x ∈ U ，我们将 C ˜ 分成两部分 ⊤ x = { C ˜ i ∈ C ˜ : C ˜ i ( x ) ≥ β } 和 ⊥ x = { C ˜ i ∈ C ˜ : C ˜ i ( x ) β } . 为了从积极方面和消极方面全面描述 x，⊤ x 和 ⊥ x 两部分都很重要，尤其是因此，在本文中，基于这两部分，我们定义了第一对 β-模糊覆盖粗糙集模型 [ ( P − ˜ , P + ˜ ) , ( Q − ˜ , Q+~)]和第二对模糊β覆盖粗糙集模型[(P-‾,P+‾),(Q-‾,Q+‾)]。这两种耦合模糊β覆盖粗糙集模型都是覆盖粗糙集理论中定义的孪生近似的推广。由于这些模型中的每一对算子不仅密切相关而且相辅相成，因此可以从正反两方面分析和解决实际问题，从而做出关键决策。那么我们接下来举几个例子来说明它们的实用价值。研究了我们的模型与之前文献中介绍的一些其他模型之间的关系，并给出了矩阵方法来计算相关的近似值并描述每对模型之间的关系。将耦合模糊β覆盖粗糙集模型推广到CCD点阵有点复杂，因为点阵中的任何两个元素通常不能相互比较。经过一番努力，我们成功地构建了L-fuzzy粗糙集理论中的理想耦合L-fuzzy β-covering粗糙集模型，它是两种耦合模糊β-covering粗糙集模型的推广。我们还获得了晶格矩阵表示来计算相关的近似值。