Endomorphisms of algebras of sequences
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.
scientific article; zbMATH DE number 5543188