Tolerances Induced by Irredundant Coverings

: Jarvinen J, Radeleczki S

Publisher: IOS PRESS

: 2015

: Fundamenta Informaticae

: FUNDAMENTA INFORMATICAE

: FUND INFORM

: 137

: 3

: 341

: 353

: 13

: 0169-2968

DOI: https://doi.org/10.3233/FI-2015-1183

: https://arxiv.org/abs/1404.5184

In this paper, we consider tolerances induced by irredundant coverings. Each tolerance R on U determines a quasiorder less than or similar to(R) by setting x less than or similar to(R) y if and only if R(x) subset of 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, less than or similar to(R)) is bounded by minimal elements and the tolerance R coincides with the product greater than or similar to(R) circle less than or similar to(R). We also show that in such a case H = {up arrow m | m is minimal in (U, less than or similar to(R))}, and for each minimal m, we have R(m) = up arrow 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.