Determination of \(\text{msd}(L^n)\) (Q1283456)

A median algebra is a set \(M\) with an operator \(m: M^3\to M\) satisfying the following axioms: \(m(a,a,b)= a\), \(m(a,b,c)= m(d,e,f)\) for each permutation \((d,e,f)\) of \((a,b,c)\) and \(m(a, m(b,c,d),c)= m(m(a,b,c), d,c)\). 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. With computer assistance, the author found that the msd of the lattice \(\{-1,0,1\}^4\) equals 2. This value is of critical importance to determine the msd of \(\{-1,0,1\}^n\) for all \(n\geq 5\) and to determine the msd of the free median algebra \(\lambda(r)\), for \(r\) a natural number \(\geq 5\) (\(\lambda(r)\) is the median stabilization of an \(r\)-point set \(S\)).