Representation of Nelson Algebras by Rough Sets Determined by Quasiorders

Details Representation of Nelson Algebras by Rough Sets Determined by Quasiorders Representation of Nelson Algebras by Rough Sets Determined by Quasiorders

2010-07-07

arxiv.org/abs/1007.1199v3

Mathematics

Representation Theory

16 pages

Scientific paper

10.1007/s00012-011-0149-9

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. We note 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 Nelson algebra defined on an algebraic lattice, then there exists a set $U$ and a quasiorder $R$ on $U$ such that $\mathbb{A} \cong \mathbb{RS}$.

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