A1 Refereed original research article in a scientific journal
Tolerances Induced by Irredundant Coverings
Authors: Jarvinen J, Radeleczki S
Publisher: IOS PRESS
Publication year: 2015
Journal: Fundamenta Informaticae
Journal name in source: FUNDAMENTA INFORMATICAE
Journal acronym: FUND INFORM
Volume: 137
Issue: 3
First page : 341
Last page: 353
Number of pages: 13
ISSN: 0169-2968
DOI: https://doi.org/10.3233/FI-2015-1183(external)
Self-archived copy’s web address: https://arxiv.org/abs/1404.5184(external)
Abstract
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.
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.