Hereditary rigidity, separation and density In memory of Professor I.G. Rosenberg

TL;DR

This paper investigates the properties of hereditarily rigid relations, especially focusing on the minimal number of linear orders needed to separate pairs of elements in finite and infinite sets, connecting the problem to set theory and topology.

Contribution

It establishes bounds and equalities for the minimal number of linear orders required for separation, linking the problem to the Generalized Continuum Hypothesis and topological density.

Findings

For finite sets, the minimal number of linear orders is bounded by a logarithmic function.

For infinite sets, the minimal number equals aleph-null under certain conditions.

The problem relates to set theory hypotheses like the GCH and topological properties of linear orders.

Abstract

We continue the investigation of systems of hereditarily rigid relations started in Couceiro, Haddad, Pouzet and Sch\"olzel [1]. We observe that on a set $V$ with $m$ elements, there is a hereditarily rigid set $\mathcal R$ made of $n$ tournaments if and only if $m(m-1)\leq 2^n$. We ask if the same inequality holds when the tournaments are replaced by linear orders. This problem has an equivalent formulation in terms of separation of linear orders. Let $h_{\rm Lin}(m)$ be the least cardinal $n$ such that there is a family $\mathcal R$ of $n$ linear orders on an $m$-element set $V$ such that any two distinct ordered pairs of distinct elements of $V$ are separated by some member of $\mathcal R$, then $ \lceil \log_2 (m(m-1))\rceil\leq h_{\rm Lin}(m)$ with equality if $m\leq 7$. We ask whether the equality holds for every $m$. We prove that $h_{\rm Lin}(m+1)\leq h_{\rm Lin}(m)+1$. If $V$ is infinite, we show that $h_{\rm Lin}(m)= \aleph_0$ for $m\leq 2^{\aleph_0}$. More generally, we prove that the two equalities $h_{\rm Lin}(m)= log_2 (m)= d({\rm Lin}(V))$ hold, where $\log_2 (m)$ is the least cardinal $\mu$ such that $m\leq 2^\mu$, and $d({\rm Lin}(V))$ is the topological density of the set ${\rm Lin}(V)$ of linear orders on $V$ (viewed as a subset of the power set $\mathcal{P}(V\times V)$ equipped with the product topology). These equalities follow from the {\it Generalized Continuum Hypothesis}, but we do not know whether they hold without any set theoretical hypothesis.