On uniformly repetitive semigroups
Let \(A\) be an alphabet, \(S\) a semigroup, \(\phi : A^ + \to S\) a morphism from the free semigroup \(A^ +\) and \(w \in A^ +\). If \(w\) has a factorization \(xw_ 1 w_ 2 \dots w_ k y\) with \(w_ i \in A^ +\), \(x,y \in A^ +\) (the free monoid on \(A\)) with \(\phi(w_ 1) = \phi(w_ 2) = \dots = \phi(w_ k)\) we say that \(w\) contains a \(k\)-power modulo \(\phi\). If, moreover, all the words \(w_ i\) are equal in length, we say that \(w\) contains a uniform \(k\)-power modulo \(\phi\). The morphism \(\phi\) is called \(k\)-repetitive (resp. uniformly \(k\)-repetitive) if there exists an integer \(R(\phi,k)\) such that each word \(w\) of \(A^ +\) of length \(R(\phi,k)\) contains a \(k\)-power modulo \(\phi\) (resp. a uniform \(k\)-power modulo \(\phi\)). We say that \(S\) is \(k\)-repetitive (resp. uniformly \(k\)- repetitive) if for each finite alphabet \(A\), each morphism \(\phi: A^ + \to S\) is \(k\)-repetitive (resp. uniformly \(k\)-repetitive). It is proved that if \(\mathbb{N}^ +\), (the non-negative integers under addition) is not uniformly \(k\)-repetitive then any finitely generated and uniformly \(k\)- repetitive semigroup is finite. (It is known that \(\mathbb{N}^ +\) is not uniformly 4-repetitive.) A bibliography of 19 papers on repetitive semigroups is listed.
scientific article; zbMATH DE number 611121
scientific article