Tolerances induced by irredundant coverings

Abstract

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

12 pages, 2 figures

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