The zero divisor conjecture and Mealy automata

2024/02/13 by Ievgen Bondarenko, Bondarenko, Ievgen, Kate Juschenko +1
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #semigroups and automata theory #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2402.08625

Abstract

The zero divisor conjecture is sufficient to prove for certain class of finitely presented groups where the relations are given by a pairing of generators. We associate Mealy automata to such pairings, and prove that the zero divisor conjecture holds for groups corresponding to invertible automata with three states. In particular, there cannot be zero divisors of support three corresponding to invertible pairings.

Related

• A connected 3-state reversible Mealy automaton cannot generate an infinite Burnside group
• Connected reversible Mealy automata of prime size cannot generate infinite Burnside groups
• A characterization of those automata that structurally generate finite groups
• The conjugacy problem in automaton groups is not solvable
• On a series of finite automata defining free transformation groups