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Tolerances Induced by Irredundant Coverings

Järvinen J., Radeleczki S.

Fundamenta Informaticae

2015

Vol. 137, nr 3

341--353

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.