On uniformly continuous functions for some profinite topologies
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.
scientific article; zbMATH DE number 6666080
scientific article