Tolerances induced by irredundant coverings

TL;DR

This paper characterizes tolerances induced by irredundant coverings using quasiorders, showing their structure and conditions for equivalence with blocks, with implications for understanding tolerance-based coverings.

Contribution

It provides a complete characterization of tolerances from irredundant coverings via quasiorders and introduces conditions for their blocks to form the covering.

Findings

Irredundant coverings correspond to quasiorders bounded by minimal elements.

The tolerance coincides with the product of the quasiorder and its inverse.

Conditions are given for the covering to match the set of R-blocks.

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$ is minimal in $(U,\lesssim_R)$} \}$, 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.