On the lattice of fuzzy rough sets

TL;DR

This paper investigates the structure of fuzzy rough sets using fuzzy approximations, characterizing when they form lattices under specific t-norms and implicators, with results applicable to finite universes and chains.

Contribution

It provides a characterization of fuzzy rough sets with respect to t-similarity relations and establishes conditions for their lattice structure under certain t-norms and implicators.

Findings

Fuzzy rough sets form lattices under specific conditions.

Complete lattice structure is achieved with min t-norm and certain implicators.

Results apply to finite universes and fixed finite chains.

Abstract

By the means of lower and upper fuzzy approximations we define quasiorders. Their properties are used to prove our main results. First, we characterize those pairs of fuzzy sets which form fuzzy rough sets w.r.t. a t-similarity relation $\theta$ on $U$, for certain t-norms and implicators. Then we establish conditions under which fuzzy rough sets form lattices. We show that for the $\min$ t-norm and any S-implicator defined by the $\max$ co-norm with an involutive negator, the fuzzy rough sets form a complete lattice, whenever $U$ is finite or the range of $\theta$ and of the fuzzy sets is a fixed finite chain.