论文标题
关于模糊粗糙集的晶格的注释，带有清晰的参考集
Notes on the lattice of fuzzy rough sets with crisp reference sets
论文作者
论文摘要
由于Zdzislaw Pawlak引入了粗糙集理论，因此提出了几种方法将粗糙集理论与模糊集理论相结合。在本文中，我们从晶格理论的角度研究了这些方法之一，即带有清晰参考集的模糊粗糙集。我们将模糊关系$ r $的下部和上部近似与核心的近似值和$ r $的支持联系起来。我们还表明，模糊粗糙的晶格对应于模糊等价关系$ r $，并且其宇宙的酥脆子集对（crisp）等价关系$ e $的粗糙套件是同构的，其中$ e $是$ r $的核心。我们建立了$ r $的确切（模糊）集与$ r $的确切（酥脆）集合之间的连接。
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 with crisp reference 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 $R$.
