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The zero divisor conjecture and Mealy automata

Ievgen Bondarenko, Kate Juschenko

TL;DR

It is proved that the zero divisor conjecture holds for groups corresponding to invertible automata with three states, and there cannot be zero divisors of support three corresponding to invertible pairings.

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.

The zero divisor conjecture and Mealy automata

TL;DR

It is proved that the zero divisor conjecture holds for groups corresponding to invertible automata with three states, and there cannot be zero divisors of support three corresponding to invertible pairings.

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.
Paper Structure (2 sections, 4 theorems, 22 equations, 1 figure)

This paper contains 2 sections, 4 theorems, 22 equations, 1 figure.

Table of Contents

  1. Mealy automata and pairing matrices
  2. The zero divisor conjecture for invertible pairings

Key Result

Proposition 1

Let $C$ be a pairing matrix and $\mathsf{A}_C$ the associated automaton. The following statements are equivalent:

Figures (1)

  • Figure 1: Example of a pairing matrix $C$ and the associated automaton $\mathsf{A}_C$

Theorems & Definitions (14)

  • Conjecture 1: Zero Divisor Conjecture
  • Conjecture 2: Direct Finiteness Conjecture
  • Conjecture 3
  • Proposition 1
  • proof
  • Definition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 4 more