Approximations Induced by Tolerance Relations

Abstract

In this paper, we examined by using Formal Concept Analysis methods, the interrelation between the lattices of upper (lower) approximations induced by two tolerance relations$$R\subseteq \rho \subseteq U\times U $$. These lattices are isomorphic (dually isomorphic) to the concept lattice$$\mathcal {L}(U,U,R^{c})$$,$$\mathcal {L}(U,U,\rho ^{c})$$ respectively, where$$R^{c}$$ and$$\rho ^{c}$$ stand for the complements of the corresponding relations. We proved sufficient conditions and we characterized the case when the concept lattice$$\mathcal {L}(U,U,\rho ^{c})$$ is a complete sublattice of$$\mathcal {L}(U,U,R^{c})$$. We used the so-called compatibility condition introduced recently and we showed that in the case when$$\rho $$ is R-compatible and$$\mathcal {L}(U,U,\rho ^{c})$$ is a complete sublattice of$$\mathcal {L} (U,U,R^{c})$$,$$\rho $$ must be an equivalence. Detailed examples for each case were presented.