Affiliations: Sirkankuja 1, 20810 Turku, Finland. [email protected] | Institute of Mathematics, University of Miskolc, 3515 Miskolc-Egyetemváros, Hungary. [email protected]
Note: [] Address for correspondence: Sirkankuja 1, 20810 Turku, Finland
Note: [] The research was carried out as a part of TAMOP-4.2.1.B-10/2/KONV-2010-0001 project, supported by the European Union, co-financed by European Social Fund.
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 º ≤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.
Keywords: Tolerance relation, Irredundant covering, Normal covering, Clique of a graph, Helly number, Rough set, Formal concept analysis
