On free semigroups of automaton transformations
type: Item
instance of: scholarly article
title: On free semigroups of automaton transformations
zbMATH Open document ID: 0920.20057
author: A. S. Olijnyk
published in: Mathematical Notes
publication date: 1999-07-21T00:00:00Z
review text: The author considers a method of constructing free semigroups of automaton transformations proposed by Grigorchuk. It is connected with results given by \textit{S. V. Aleshin} [Mat. Zametki 11, No. 3, 319-328 (1972; Zbl 0246.20024)], \textit{D. B. A. Epstein} [J. Algebra 19, 261-262 (1971; Zbl 0222.22012)], \textit{J. D. Dixon} [Bull. Lond. Math Soc. 22, No. 3, 222-226 (1990; Zbl 0675.20003)] and \textit{M. Bhattacharjee} [J. Algebra 172, No. 1, 134-146 (1995; Zbl 0920.20025)]. It is established that the subset of free \(k\)-generated subsemigroups of the semigroup of all automaton transformations is a second category set (in the sense of Baire category) in the set of all \(k\)-generated subsemigroups. In addition, a continuum series of pairs of automaton transformations generating a free semigroup of rank two is indicated, and a criterion for this semigroup to be a finite-automaton semigroup is obtained. In terms of wreath products, a special class of automaton transformations for this generation is given.
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M35
zbMATH DE Number: 1228275
zbMATH Keywords: finite automata
zbMATH Keywords: automaton transformations
zbMATH Keywords: transitive semigroups
zbMATH Keywords: first category in the sense of Baire
zbMATH Keywords: wreath products
reviewed by: Kalčo Todorov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1228275