## The equations $$h(w)=w^ n$$ in binary alphabets.(English)Zbl 0548.20044

Let h be an endomorphism on a finitely generated word semigroup $$A^*$$. A solution of the equations $$h(x)=x^ n (n=2,3,...)$$ is just a word w in $$A^*$$ for which (*) $$h(w)\in w^ 2w^*$$. The solutions (*) in a word semigroup of binary alphabets are investigated. A complete account on all the solutions and on all the morphisms possessing a solution is given. The authors point out that in the case of binary alphabets, the primitive solution w is of length at most $$\max\{| h(a)|;| h(b)|\},$$ however this is not so in alphabets of larger size.
Reviewer: K.-P.Shum

### MSC:

 20M05 Free semigroups, generators and relations, word problems 20M15 Mappings of semigroups
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### References:

 [1] Harju, T.; Linna, M., On the periodicity of morphisms on free monoids, (1984), Submitted for publication [2] Linna, M., On periodic ω-sequences obtained by iterating morphisms, Ann. univ. turkuensis, 186, I, 64-71, (1984), Ser. A · Zbl 0547.68074
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