Automaton semigroup free products revisited

An improvement on earlier results on free products of automaton semigroups; showing that a free product of two automaton semigroups is again an automaton semigroup providing there exists a homomorphism from one of the base semigroups to the other. The result is extended by induction to give a condition for a free product of finitely many automaton semigroups to be an automaton semigroup.Comment: 5 page

The Self-Similarity of Free Semigroups and Groups (Logic, Algebraic system, Language and Related Areas in Computer Science)

We give a survey on results regarding self-similar and automaton presentations of free groups and semigroups and related products. Furthermore, we discuss open problems and results with respect to algebraic decision problems in this area

Automaton Semigroups and Groups: On the Undecidability of Problems Related to Freeness and Finiteness

In this paper, we study algorithmic problems for automaton semigroups and automaton groups related to freeness and finiteness. In the course of this study, we also exhibit some connections between the algebraic structure of automaton (semi)groups and their dynamics on the boundary. First, we show that it is undecidable to check whether the group generated by a given invertible automaton has a positive relation, i.e. a relation p = 1 such that p only contains positive generators. Besides its obvious relation to the freeness of the group, the absence of positive relations has previously been studied and is connected to the triviality of some stabilizers of the boundary. We show that the emptiness of the set of positive relations is equivalent to the dynamical property that all (directed positive) orbital graphs centered at non-singular points are acyclic. Gillibert showed that the finiteness problem for automaton semigroups is undecidable. In the second part of the paper, we show that this undecidability result also holds if the input is restricted to be bi-reversible and invertible (but, in general, not complete). As an immediate consequence, we obtain that the finiteness problem for automaton subsemigroups of semigroups generated by invertible, yet partial automata, so called automaton-inverse semigroups, is also undecidable. Erratum: Contrary to a statement in a previous version of the paper, our approach does not show that that the freeness problem for automaton semigroups is undecidable. We discuss this in an erratum at the end of the paper

A characterization of those automata that structurally generate finite groups

Antonenko and Russyev independently have shown that any Mealy automaton with no cycles with exit--that is, where every cycle in the underlying directed graph is a sink component--generates a fi- nite (semi)group, regardless of the choice of the production functions. Antonenko has proved that this constitutes a characterization in the non-invertible case and asked for the invertible case, which is proved in this paper

Some undecidability results for asynchronous transducers and the Brin-Thompson group 2V

Using a result of Kari and Ollinger, we prove that the torsion problem for elements of the Brin-Thompson group 2V is undecidable. As a result, we show that there does not exist an algorithm to determine whether an element of the rational group R of Grigorchuk, Nekrashevich, and Sushchanskii has finite order. A modification of the construction gives other undecidability results about the dynamics of the action of elements of 2V on Cantor Space. Arzhantseva, Lafont, and Minasyanin prove in 2012 that there exists a finitely presented group with solvable word problem and unsolvable torsion problem. To our knowledge, 2V furnishes the first concrete example of such a group, and gives an example of a direct undecidability result in the extended family of R. Thompson type groups.Comment: 16 pages, 3 figure

Some undecidability results for asynchronous transducers and the Brin-Thompson group 2V

Funding: partial support by UK EPSRC grant EP/H011978/1 (CB).Using a result of Kari and Ollinger, we prove that the torsion problem for elements of the Brin-Thompson group 2V is undecidable. As a result, we show that there does not exist an algorithm to determine whether an element of the rational group R of Grigorchuk, Nekrashevich, and Sushchanskiĭ has finite order. A modification of the construction gives other undecidability results about the dynamics of the action of elements of 2V on Cantor space. Arzhantseva, Lafont, and Minasyanin proved in 2012 that there exists a finitely presented group with solvable word problem and unsolvable torsion problem. To our knowledge, 2V furnishes the first concrete example of such a group and gives an example of a direct undecidability result in the extended family of R. Thompson type groups.PostprintPeer reviewe