In this paper, we show that every quasiorder $R$ induces a Nelson algebra$\mathbb{RS}$ such that the underlying rough set lattice $RS$ is algebraic. Wenote that $\mathbb{RS}$ is a three-valued {\L}ukasiewicz algebra if and only if$R$ is an equivalence. Our main result says that if $\mathbb{A}$ is a Nelsonalgebra defined on an algebraic lattice, then there exists a set $U$ and aquasiorder $R$ on $U$ such that $\mathbb{A} \cong \mathbb{RS}$.
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机译:在本文中,我们证明了每个准序$ R $都诱导了一个Nelson代数$ \ mathbb {RS} $,使得基础粗糙集格$ RS $是代数的。我们注意到,当且仅当$ R $是等价的时,$ \ mathbb {RS} $是三值{\ L} ukasiewicz代数。我们的主要结果表明,如果$ \ mathbb {A} $是在代数格上定义的纳尔逊代数,则在$ U $上存在一组$ U $和准序$ R $,使得$ \ mathbb {A} \ cong \ mathbb {RS} $。
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