On uniformly repetitive semigroups
type: Item
instance of: scholarly article
title: On uniformly repetitive semigroups
zbMATH Open document ID: 0805.20052
DOI: 10.1007/BF02573477
author: Giuseppe Pirillo
author: Stefano Varricchio
published in: Semigroup Forum
publication date: 1995-02-08T00:00:00Z
full work available at URL: https://eudml.org/doc/135340
review text: 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.
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M15
zbMATH DE Number: 611121
zbMATH Keywords: free semigroup
zbMATH Keywords: factorization
zbMATH Keywords: free monoid
zbMATH Keywords: uniform \(k\)-power
zbMATH Keywords: finite alphabet
zbMATH Keywords: uniformly \(k\)-repetitive semigroup
zbMATH Keywords: repetitive semigroups
reviewed by: Peter M. Higgins
MaRDI profile type: Publication
cites work: An interesting combinatorial method in the theory of locally finite semigroups
cites work: Strongly non-repetitive sequences and progression-free sets
cites work: Q5580276
cites work: Sur une construction de Bruck et Reilly. (On a construction of Bruck and Reilly.)
cites work: Q5622446
cites work: Generalisation du theoreme de van der Waerden sur les semi-groupes repetitifs
cites work: Characterization of the repetitive commutative semigroups
cites work: Two combinatorial properties of partitions of the free semigroup into finitely many parts
cites work: On a natural extension of Jacob's ranks
cites work: Q3659988
cites work: Q5646912
cites work: Q4006441
cites work: Properties of integers and finiteness conditions for semigroups
OpenAlex ID: W2082384497
description: scientific article; zbMATH DE number 611121