Words guaranteeing minimum image

Given a positive integer n and a finite alphabet Σ, a word w over Σ is said to guarantee minimum image if, for every homomorphism φ from the free monoid Σ* over Σ into the monoid of all transformations of an n-element set, the range of the transformation wφ has the minimum cardinality among the ranges of all transformations of the form vφ where v runs over Σ*. Although the existence of words guaranteeing minimum image is pretty obvious, the problem of their explicit description is very far from being trivial. Sauer and Stone in 1991 gave a recursive construction for such a word w but the length of their word was doubly exponential (as a function of n). We first show that some known results of automata theory immediately lead to an alternative construction that yields a simpler word that guarantees minimum image: it has exponential length, more precisely, its length is O(|Σ|(n³-n)). Then with some more effort, we find a word guaranteeing minimum image similar to that of Sauer and Stone but of length O(|Σ|(n²-n)). On the other hand, we prove that the length of any word guaranteeing minimum image cannot be less than |Σ|^n-1.