Endomorphisms of algebras of sequences
type: Item
instance of: scholarly article
title: Endomorphisms of algebras of sequences
zbMATH Open document ID: 1211.11014
author: Abdelkader Necer
author: Ahmed Ait Mokhtar
author: Alain Salinier
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-04-14T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10828
review text: Let \(K\) be a commutative ring with unit and \(S(K)=\{u(i);~i\in\mathbb{N}, u(i)\in K\}\). Then \(S(K)\) is a commutative ring for the natural addition and the Hadamard product. Put the discrete topology on \(K\) and the Cartesian topology on \(S(K)\). The goal of this paper is the study of the continuous endomorphisms of \(S(K)\). Let \(T\) be the shift defined on \(S(K)\) by \(\forall u\in S(K)\), \(\forall n\in\mathbb{N}\), \((Tu)(n)=u(n+1)\). Since \(T\) is \(K-\)linear then \(S(K)\) is a \(K[T]-\)module. A sequence \(u\) is said to be recognisable if it satisfies one of the following equivalent conditions: (i) The \(K[T]-\)module of \(S(K)\) generated by \(u\) is of finite type over \(K\). (ii) The generating series \(\displaystyle f_u=\sum_{n:0}^{\infty}u(n)X^n\) is a rational function whose denominator is a polynomial with coefficients in \(K\) and constant coefficient invertible in \(K\). (iii) \(u\) satisfies a linear recurrence \(\displaystyle\sum_{i:0}^{l-1}p_iu(n+i)+u(n+h)=0\), for \(n\geq 0\) with coefficients \(p_0,\ldots,p_{h-1}\in K\). Let \(r(K)\) be the set of all the recognisable sequences. A special attention is given to \(r(K)\) in this paper. Suppose that \(K\) is a field of zero characteristic. Let \((d,t)\in\mathbb{N}^*\times\mathbb{N}\), \(a=(a_i)_{0\leq i<d}\in\mathbb{N}^d\), \(b=(b_i)_{0\leq i<d}\in\mathbb{Z}^d\) and \(c=(c_l)_{0\leq l<dt}\in\mathbb{N}^{dt}\) such that \(a_it+b_i\geq 0\) for all \(i\in\{0,1,\ldots,d-1\}\). Define the map \(\phi_{(d,a,b,c,t)}:\mathbb{N}\longrightarrow\mathbb{N}\) by \(\phi_{(d,a,b,c,t)}(n)=c_n\) if \(n<dt\) and \(\phi_{(d,a,b,c,t)}(n)=a_r[{n\over d}]+b_r\) if \(n>dt\); where \(r=d\{{n\over d}\}\). Such applications \(\phi_{(d,a,b,c,t)}\) are called semi-affine. The authors prove that every continuous endomorphism \(f\) of \(r(K)\) has the shape \(f(u)=u\circ\phi\) where \(\phi\) is a semi-affine map.
reviewed by: Ali Benhissi
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 5543188
zbMATH Keywords: Recognisable sequence
zbMATH Keywords: Rational function
zbMATH Keywords: Linear recurrent sequence
zbMATH Keywords: Semi-affine map
zbMATH Keywords: Continuous endomorphism
MaRDI profile type: Publication
OpenAlex ID: W1485355966
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DOI: 10.5802/JTNB.613
description: scientific article; zbMATH DE number 5543188