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On (⊙, &)-fuzzy rough sets based on residuated and co-residuated lattices

机译：基于残差和共残差格的（⊙，＆）-模糊粗糙集

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Rough sets were introduced by Pawlak as a formal tool for dealing with imprecision and uncertainty in data analysis. After that time, varieties of fuzzy generalizations of rough approximations have been investigated in the literature. In particular, generalized fuzzy rough sets determined by a triangular norm have been considered. On the other hand, binary operations & and circle dot on complete residuated and co-residuated lattice L can be regarded as two more extensive operations than left continuous triangular norms and right continuous triangular conorms, respectively. Therefore, as a further generalization of the notion of rough sets, this paper is devoted to proposing (circle dot, &)-fuzzy rough sets based on residuated and co-residuated lattices from both constructive and axiomatic approaches. In the constructive approach, we define a pair of lower and upper L-fuzzy rough approximation operators determined by circle dot and &, respectively. Meanwhile, various classes of (circle dot, &)-fuzzy rough sets are investigated. In the axiomatic approach, the axiomatic characterizations of different (circle dot, &)-fuzzy rough approximation operators are studied. Furthermore, the topological properties of (circle dot, &)-fuzzy rough sets are discussed, especially from the categorical point of view. At the end of this paper, we give a brief introduction to a new model of fuzzy rough sets and discuss some properties of it. (c) 2017 Elsevier B.V. All rights reserved.
机译：Pawlak引入了粗糙集作为处理数据分析中不精确性和不确定性的正式工具。在那之后，文献中已经研究了各种粗糙近似的模糊概括。特别地，已经考虑了由三角模确定的广义模糊粗糙集。另一方面，完全残差和共残差的格L上的二元运算＆和圆点分别可以看作是比左连续三角形三角范式和右连续三角形三角范式更广泛的两种运算。因此，作为粗糙集概念的进一步概括，本文致力于基于构造方法和公理方法的残差和共残差格点，提出（圆点和模糊）粗糙集。在构造方法中，我们定义了分别由圆点和＆决定的一对上下模糊粗略近似算子。同时，研究了各种类别的（圆点和＆）模糊粗糙集。在公理方法中，研究了不同（圆点和模糊）粗糙近似算子的公理表征。此外，特别从分类的角度，讨论了（圆点和）模糊粗糙集的拓扑性质。在本文的最后，我们简要介绍了一种新的模糊粗糙集模型并讨论了它的一些特性。 （c）2017 Elsevier B.V.保留所有权利。

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