Related papers: Groups Synchronizing a Transformation of Non-Unifo…
We develop an effective and natural approach to interpret any semigroup admitting a special language of greedy normal forms as an automaton semigroup,namely the semigroup generated by a Mealy automaton encoding the behaviour of such a…
Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabiliser in $G$ is trivial. In this paper we introduce and study an associated graph $\Sigma(G)$, which we call the Saxl graph of…
Let us say that a Cayley graph $\Gamma$ of a group $G$ of order $n$ is a Cerny Cayley graph if every synchronizing automaton containing $\Gamma$ as a subgraph with the same vertex set admits a synchronizing word of length at most $(n-1)^2$.…
A detailed proof is given of a theorem describing the centraliser of a transitive permutation group, with applications to automorphism groups of objects in various categories of maps, hypermaps, dessins, polytopes and covering spaces, where…
This paper introduces and studies a notion of \emph{algorithmic randomness} for subgroups of rationals. Given a randomly generated additive subgroup $(G,+)$ of rationals, two main questions are addressed: first, what are the model-theoretic…
Let $K$ be a field and $f:\mathbb{P}^N \to \mathbb{P}^N$ a morphism. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group $\text{PGL}_{N+1}$. The group of automorphisms, or…
This paper investigates the finite generation of cluster automorphism groups. By applying the pseudo $\mathbb{N}$-grading introduced in our previous work, we establish a sufficient condition for a cluster automorphism group to be finitely…
This paper introduces the notion of orbit coherence in a permutation group. Let $G$ be a group of permutations of a set $\Omega$. Let $\pi(G)$ be the set of partitions of $\Omega$ which arise as the orbit partition of an element of $G$. The…
The operation of switching a graph $\Gamma$ with respect to a subset $X$ of the vertex set interchanges edges and non-edges between $X$ and its complement, leaving the rest of the graph unchanged. This is an equivalence relation on the set…
A word is called carefully synchronising for a partial deterministic finite semi-automaton if it maps all states to the same state. Equivalently, it is a composition of partial transformations equal to a constant total transformation. There…
A generalized numerical semigroup is a submonoid $S$ of $\mathbb{N}^d$ with finite complement in it. We characterize isomorphisms between these monoids in terms of permutation of coordinates. Considering the equivalence relation that…
We consider the structures given by repeatedly generalising the definition of finite state automata by symmetry considerations, and constructing analogues of transition monoids at each step. This approach first gives us non-deterministic…
We study a connection between synchronizing automata and its set $M$ of minimal reset words, i.e., such that no proper factor is a reset word. We first show that any synchronizing automaton having the set of minimal reset words whose set of…
We investigate the intersection problem for finite semigroups, which asks for a given set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. We…
In this work we introduce the notion of almost-symmetry for generalized numerical semigroups. In addition to the main properties occurring in this new class, we present several characterizations for its elements. In particular we show that…
A synchronizing word of a deterministic finite complete automaton is a word whose action maps every state to a single one. Finding a shortest or a short synchronizing word is a central computational problem in the theory of synchronizing…
We present the ideas behind an algorithm to compute normalizers of primitive groups with non-regular socle in polynomial time. We highlight a concept we developed called permutation morphisms and present timings for a partial implementation…
We introduce the Hermitian-invariant group $\Gamma_f$ of a proper rational map $f$ between the unit ball in complex Euclidean space and a generalized ball in a space of typically higher dimension. We use properties of the groups to define…
Constant Rank (CR) state machines play an important role in the general structure theory of Finite State Machines. A machine is of constant rank if each input and input-sequence maps the state set onto the same number of next states.…
Resetting a system's state plays a fundamental role in physics, engineering, computer science, and many other fields. Here we focus on a method originally proposed in automata theory. The state of an automaton evolves according to a set of…