In search of a word with special combinatorial properties (Q2702051)

Let \(\alpha: A^{+}\to B^{+}\) be a semigroup homomorphism. We say that \(\alpha\) is cyclic if there exists a word \(b\in B^{+}\) with \(Im(\alpha)\subseteq b^{+}\). The rank of \(\alpha\) is the size of a base of the least subsemigroup of \(B^{+}\) containing \(Im(\alpha)\) and isomorphic with a free semigroup. If \(e=e'\) is a semigroup word equation over an alphabet \(X\) then a semigroup homomorphism \(\alpha :X^{+}\to A^{+}\) is called a solution of \(e=e'\) whenever \(\alpha (e)=\alpha (e')\). If \(\beta :X^{+}\to B^{+}\) is another solution of \( e=e'\) such that \(\beta =\theta\circ\alpha\) for a semigroup homomorphism \(\theta :A^{+}\to B^{+}\), then we write \(\alpha \leq\beta\). The relation \( \leq\) is a partial ordering on the set of all solutions of \(e=e'\). For \(n>2\) let \((*)\) denote the family of equations \((x_1x_2\dots x_n)^2=x_1^2x_2^2\dots x_n^2\) and \((x_1x_2\dots x_n)^3=x_1^3x_2^3\dots x_n^3\). It is proved that if \(\alpha :X^{+}\to A^{+}\) is a non-cyclic solution of \((*)\) such that for any non-cyclic solution \(\alpha':X^{+}\to A^{+}\) of \((*)\) we have \(|\alpha (x_1x_2\dots x_n)|\leq |\alpha'(x_1x_2\dots x_n)|\), then the rank of \( \alpha\) is 2 and \(\alpha\) is minimal with respect to \(\leq\). Moreover, if \(A=\{a,b\}\) then \(\alpha (x_i)\) contains both letters \(a\) and \(b\) for all \(i=1,2,\dots,n\). A generalization of this result for a system of equations with general exponents \(r\) and \(s\) (instead of \(2\) and \(3)\) is given without proofs.NEWLINENEWLINEFor the entire collection see [Zbl 0940.00028].