Hereditary rigidity, separation and density In memory of Professor I.G. Rosenberg
share
We continue the investigation of systems of hereditarily rigid relations started in Couceiro, Haddad, Pouzet and Schölzel [1]. We observe that on a set V with m elements, there is a hereditarily rigid set ℛ made of n tournaments if and only if m(m-1)≤ 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_ Lin(m) be the least cardinal n such that there is a family 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 ℛ, then ⌈log_2 (m(m-1))⌉≤ h_ Lin(m) with equality if m≤ 7. We ask whether the equality holds for every m. We prove that h_ Lin(m+1)≤ h_ Lin(m)+1. If V is infinite, we show that h_ Lin(m)= ℵ_0 for m≤ 2^ℵ_0. More generally, we prove that the two equalities h_ Lin(m)= log_2 (m)= d( Lin(V)) hold, where log_2 (m) is the least cardinal μ such that m≤ 2^μ, and d( Lin(V)) is the topological density of the set Lin(V) of linear orders on V (viewed as a subset of the power set 𝒫(V× V) equipped with the product topology). These equalities follow from the Generalized Continuum Hypothesis, but we do not know whether they hold without any set theoretical hypothesis.
READ FULL TEXT