The median stabilization degree of a median algebra (Q1283450)

A median algebra consists of a set \(M\) and a median operator on \(M\), which is a symmetric function \(m: M^3\to M\) such that \(m(a,a,b)= a\) and \(m(m(a,b,c),d,c)= m(a, m(b,c,d), c)\). The median stabilization degree (msd, for short) of a median algebra measures the largest possible number of steps needed to generate a subalgebra with an arbitrary set of generators. In this paper, the authors determine the value of the msd of a graphic \(n\)-cube \(Q_n\) and derive an estimation of the msd for the natural median operator of \(\mathbb{R}^n\) which is sharp up to one or two units. Interestingly, the msd of \(Q_n\) and \(\mathbb{R}^n\) grows like \(\log_{15}n\). Finally, the authors characterize median algebras and median graphs of \(\text{msd}\leq 1\) in terms of forbidden subspaces.