On uniformly continuous functions for some profinite topologies
type: Item
instance of: scholarly article
title: On uniformly continuous functions for some profinite topologies
zbMATH Open document ID: 1368.20072
author: Pedro V. Silva
author: Jean-Éric Pin
published in: Theoretical Computer Science
publication date: 2016-12-19T00:00:00Z
review text: For a pseudovariety \(\mathbf V\) of finite monoids, a recognizable subset \(S\) of a monoid \(M\) is said to be a \(\mathbf V\)-subset if the syntactic monoid of \(S\) lies in \(\mathbf V\). A function \(f: M\to N\) is \(\mathbf V\)-preserving if \(f^{-1}(T)\) is a \(\mathbf V\)-subset of \(M\) for each \(\mathbf V\)-subset \(T\subseteq N\), and \(f\) is hereditary \(\mathbf V\)-preserving if it is \(\mathbf W\)-preserving for every subpseudovariety \({\mathbf W}\subseteq\mathbf V\). The authors characterize hereditary \(\mathbf V\)-preserving functions from a free finitely generated commutative monoid to the additive monoid of non-negative integers for each of the following pseudovarieties: the pseudovariety \({\mathbf G}_p\) of all finite \(p\)-groups, where \(p\) is a prime number (Theorem~3.12), the pseudovariety \(\mathbf G\) of all finite groups (Theorem~4.2), the pseudovariety of all finite aperiodic monoids (Theorem~5.4), and the pseudovariety of all finite monoids (Theorem~7.2). For \({\mathbf V}={\mathbf G}_p\) or \({\mathbf V}=\mathbf G\), the authors also characterize hereditary \(\mathbf V\)-preserving functions to the group of integers from each of the following monoids: a free finitely generated commutative monoid, a free finitely generated monoid, a free finitely generated abelian group. The characterizations of hereditary \({\mathbf G}_p\)-preserving or \(\mathbf G\)-preserving functions are given in terms of their Mahler expansions and look quite elegant.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M15
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 68Q70
zbMATH DE Number: 6666080
zbMATH Keywords: profinite topology
zbMATH Keywords: regularity preserving
zbMATH Keywords: regular language
zbMATH Keywords: variety of finite monoids
zbMATH Keywords: \(p\)-adic
MaRDI profile type: Publication
full work available at URL: j.tcs.2016.06.013
OpenAlex ID: W2437272661
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DOI: 10.1016/J.TCS.2016.06.013
description: scientific article; zbMATH DE number 6666080