On free semigroups of automaton transformations (Q1272282)

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.