Finitely generated clones of terms (Q1119672)

If A is an algebra, I(A) is the clone of its term functions. Let \(d\geq 2\). By a near-unanimous sequence of length \(d+1\) we mean a sequence \((x_ 0,...,x_ d)\) such that there is an i, \(0\leq i\leq d\), with \(x_ j=x_ k\) for all j,k\(\neq i\). For any algebra A set \(\lambda (A)=\inf (\lambda |\) I(A) is generated by its terms of arity \(\leq \lambda)\). For any natural numbers n and d with \(d\geq 2\) set \(\lambda_ d(n)=\sup (\lambda (A)|\) \(| A| =n\) and A has \((d+1)\)-ary near-unanimity term). It was already known that the clone of terms of any n-element algebra that has a \((d+1)\)-ary near-unanimity term is generated by the terms of arity \(\lambda_ d(n)\). In this paper the author investigates the magnitude of \(\lambda_ d(n)\). In fact he proves: Theorem 1. For each pair of integers \(n\geq 3\) and \(d\geq 2\), \((n+d-2)(n-2)^{d-1}\leq \lambda_ d(n)\leq n^ d-1.\) Theorem 2. If \(n\geq 5\), \(\lambda_ 2(n)=n(n-2).\) He concludes stating the following problem. Find a much sharper upper bound on \(\lambda_ d(n)\) for \(d>2\). Indeed, is \(\lambda_ d(n)=(n+d-2)(n-2)^{d-1}\) for sufficiently large n?