Related papers: Decidability problems in automaton semigroups
We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc), which satisfy some natural…
We introduce the notion of a subgraph generated by an $R$-word $r$ of the Sch\"{u}tzenberger graph of a positive word $w$, $S\Gamma(w)$, where $w$ contains $r$ as its subword. We show that the word problem for a finitely presented Adian…
We consider blind, deterministic, finite automata equipped with a register which stores an element of a given monoid, and which is modified by right multiplication by monoid elements. We show that, for monoids M drawn from a large class…
The Tits alternative states that a finitely generated matrix group either contains a nonabelian free subgroup $F_2$, or it is virtually solvable. This paper considers two decision problems in virtually solvable matrix groups: the Identity…
We prove that a semigroup generated by a reversible two-state Mealy automaton is either finite or free of rank 2. This fact leads to the decidability of finiteness for groups generated by two-state or two-letter invertible-reversible Mealy…
We examine the computational complexity of problems in which we are given generators for a partial bijection semigroup and asked to check properties of the generated semigroup. We prove that the following problems are in AC$^0$: (1)…
Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of…
Given a finite group $G$, the Engel graph of $G$ is a directed graph encoding pairs of elements satisfying some Engel word. From the work of Detomi, Lucchini and Nemmi, the strongly connectivity of the Engel graph of an arbitrary group $G$…
With each semigroup one can associate a partial algebra, called the biordered set, which captures important algebraic and geometric features of the structure of idempotents of that semigroup. For a biordered set $\mathcal{E}$, one can…
The Krohn-Rhodes Theorem proves that a finite semigroup divides a wreath product of groups and aperiodic semigroups. Krohn-Rhodes complexity equals the minimal number of groups that are needed. Determining an algorithm to compute complexity…
We consider pairs of finitely presented, residually finite groups $P\hookrightarrow\G$ for which the induced map of profinite completions $\hat P\to \hat\G$ is an isomorphism. We prove that there is no algorithm that, given an arbitrary…
A left Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is a left Engel element precisely…
The knapsack problem is a classic optimisation problem that has been recently extended in the setting of groups. Its study reveals to be interesting since it provides many different behaviours, depending on the considered class of groups.…
A right Engel sink of an element $g$ of a group $G$ is a set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, $g$ is a right Engel element…
We consider the growth, order, and finiteness problems for automaton (semi)groups. We propose new implementations and compare them with the existing ones. As a result of extensive experimentations, we propose some conjectures on the order…
We prove that many dynamical properties of group cellular automata (i.e., cellular automata defined on any finite group and with global rule which is an endomorphism), including surjectivity, injectivity, sensitivity to initial conditions,…
A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y in G such that for any x in G the n-th commutator [x,y,...,y] equals 1 for n big enough. We obtain a…
It is known that if every group satisfying an identity of the form yx ~ xU(x,y)y is abelian, so is every semigroup that satisfies that identity. Because a group has an identity element and the cancellation property, it is easier to show…
(Free-abelian)-by-free, self-similar groups generated by finite self-similar sets of tree automorphisms and having unsolvable conjugacy problem are constructed. Along the way, orbit undecidable, free subgroups of GL_d(Z), for d > 5, and…
We construct an automaton group with a PSPACE-complete word problem, proving a conjecture due to Steinberg. Additionally, the constructed group has a provably more difficult, namely EXPSPACE-complete, compressed word problem and acts over a…