| Given a positive integer n and a finite alphabet A, a word w over A is said to guarantee minimal image if, for every homomorphism f from the free monoid A* over A into the monoid of all transformations of an n-element set, the range of the transformation wf has the minimum cardinality among the ranges of all transformations of the form vf where v runs over A*. Although the existence of words guaranteeing minimal 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 the word resulting from that construction was doubly exponential (as a function of n). We first show that some known results of automata theory immediately lead to an alternative construction which yields a simpler word that guarantees minimal image: it has exponential length, more precisely, its length is O(|A|^(n^3-n)). Then using a different approach, we find a word guaranteeing minimal image similar to that of Sauer and Stone but of the length O(|A|^(n^2-n)). On the other hand, we observe that the length of any word guaranteeing minimal image cannot be less than |A|^(n-1). |
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