Tolerances Induced by Irredundant Coverings

Abstract

In this paper, we consider tolerances induced by irredundant coverings. Each tolerance R on U determines a quasiorder R by setting x R y if and only if R(x) ⊆ R(y). We prove that for a tolerance R induced by a covering H of U , the covering H is irredundant if and only if the quasiordered set (U, R ) is bounded by minimal elements and the tolerance R coincides with the product R • R . We also show that in such a case H = {↑m | m is minimal in (U, R )}, and for each minimal m, we have R(m) = ↑m. Additionally, this irredundant covering H inducing R consists of some blocks of the tolerance R. We give necessary and sufficient conditions under which H and the set of R-blocks coincide. These results are established by applying the notion of Helly numbers of quasiordered sets.

Representing Regular Pseudocomplemented Kleene Algebras by Tolerance-Based Rough sets

“…A [JR15]. We can now write the following lemma which states that each irredundant covering H consists of such R H (x)-sets that are blocks of R H .…”