On Sets of Words of Rank Two

Given a (finite or infinite) subset X of the free monoid \(A^*\) over a finite alphabet A, the rank of X is the minimal cardinality of a set F such that \(X \subseteq F^*\). A submonoid M generated by k elements of \(A^*\) is k-maximal if there does not exist another submonoid generated by at most k words containing M. We call a set \(X \subseteq A^*\) primitive if it is the basis of a |X|-maximal submonoid. This extends the notion of primitive word: indeed, \(\{w\}\) is a primitive set if and only if w is a primitive word. By definition, for any set X, there exists a primitive set Y such that \(X \subseteq Y^*\). The set Y is therefore called a primitive root of X. As a main result, we prove that if a set has rank 2, then it has a unique primitive root. This result cannot be extended to sets of rank larger than 2.