Since the theory of rough sets was introduced by Zdzislaw Pawlak, several approaches have been proposed to combine rough set theory with fuzzy set theory. In this paper, we examine one of these approaches, namely fuzzy rough sets, from a lattice theoretic point of view. We connect the lower and upper approximations of a fuzzy relation $R$ to the approximations of the core and support of $R$. We also show that the lattice of fuzzy rough sets corresponding to a fuzzy equivalence relation $R$ and the crisp subsets of its universe is isomorphic to the lattice of rough sets for the (crisp) equivalence relation $E$, where $E$ is the core of $R$. We establish a connection between the exact (fuzzy) sets of $R$ and the exact (crisp) sets of the support of $R$. Additionally, we examine some properties of a special case of a fuzzy relation.

中文翻译：

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带有清晰参考集的模糊粗糙集格注

自从 Zdzislaw Pawlak 引入粗糙集理论以来，已经提出了几种将粗糙集理论与模糊集理论相结合的方法。在本文中，我们从格论的角度研究了其中一种方法，即模糊粗糙集。我们将模糊关系 $R$ 的下近似和上近似与 $R$ 的核心和支持的近似联系起来。我们还表明，对应于模糊等价关系 $R$ 的模糊粗糙集格及其全域的清晰子集与（清晰）等价关系 $E$ 的粗糙集格同构，其中 $E$ 是$R$ 的核心。我们在 $R$ 的精确（模糊）集和 $R$ 支持的精确（清晰）集之间建立联系。此外，我们检查了模糊关系的特殊情况的一些属性。