Representation of Nelson Algebras by Rough Sets Determined by Quasiorders

In this paper, we show that every quasiorder $R$ induces a Nelson algebra $\mathbb{R}\mathbb{S}$ such that the underlying rough set lattice $RS$ is algebraic. We note that $\mathbb{R}\mathbb{S}$ 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 Nelson algebra defined on an algebraic lattice, then there exists a set $U$ and a quasiorder $R$ on $U$ such that $\mathbb{A}\cong \mathbb{R}\mathbb{S}$.