Related papers: The zero divisor conjecture and Mealy automata
We consider the two generalizations of lamplighter groups: automata groups generated by Cayley machine and cross-wired lamplighter groups. For a finite step two nilpotent group with central squares, we study its associated Cayley machine…
The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with $a$ and $b$ adjacent if $ab=0$. We show that the class of zero-divisor graphs is universal, in the…
We explore a natural class of semigroups that have word problem decidable by finite state automata. Among the main results are invariance of this property under change of generators, invariance under basic algebraic constructions and…
The regular separability problem asks, for two given languages, if there exists a regular language including one of them but disjoint from the other. Our main result is decidability, and PSpace-completeness, of the regular separability…
We give a presentation of various results on zero-groups in o-minimal structures together with some new observations. In particular we prove that if G is a definably connected definably compact group in an o-minimal expansion of a real…
For every non-trivial finite abelian group $A$, we exhibit a bireversible automaton generating the lamplighter group $A \wr \mathbb{Z}$.
In this paper, we study the genera of zero-divisor graphs with respect to ideals in finite rings.
In this paper we establish a formal connection between the structure of ideals in integers rings and the theory of additive combinatorics. For integers rings with cyclic class groups, we prove a structural theorem demonstrating that every…
In Section 1 of the paper, we prove McCoy's property for the zero-divisors of polynomials in semirings. We also investigate zero-divisors of semimodules and prove that under suitable conditions, the monoid semimodule $M[G]$ has very few…
We describe certain sufficient conditions for an infinitely divisible probability measure on a class of connected Lie groups to be embeddable in a continuous one-parameter convolution semigroup of probability measures. (Theorem 1.3). This…
For a commutative ring $R$ with identity, the zero-divisor graph of $R$, denoted $\Gamma(R)$, is the graph whose vertices are the non-zero zero divisors of $R$ with two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$. In…
We consider a conjecture that identifies two types of base point free divisors on $\bar{M}_{0,n}$. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated to…
Esparza and Reiter have recently conducted a systematic comparative study of weak asynchronous models of distributed computing, in which a network of identical finite-state machines acts cooperatively to decide properties of the network's…
Bell's theorem is reformulated and proved in the pure mathematical terms of automata theory, avoiding any physical or ontological notions. It is stated that no pair of finite probabilistic sequential machines can reproduce in its output the…
In 2021, J.~Agler and J.~E. McCarthy proposed a two-step programme toward the celebrated Krzy\.z conjecture. The first step is to prove an entropy conjecture for polynomials whose zeros all lie on the unit circle; the second is to establish…
Many decision problems concerning cellular automata are known to be decidable in the case of algebraic cellular automata, that is, when the state set has an algebraic structure and the automaton acts as a morphism. The most studied cases…
Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a…
We devise an algorithm which, given a bounded automaton A, decides whether the group generated by A is finite. The solution comes from a description of the infinite sequences having an infinite A-orbit using a deterministic finite-state…
A deterministic finite automaton is synchronizing if there exists a word that sends all states of the automaton to the same state. \v{C}ern\'y conjectured in 1964 that a synchronizing automaton with $n$ states has a synchronizing word of…
A meadow is a zero totalised field (0^{-1}=0), and a cancellation meadow is a meadow without proper zero divisors. In this paper we consider differential meadows, i.e., meadows equipped with differentiation operators. We give an equational…