Linear equations and recursively enumerable sets

TL;DR

This paper explores the relationship between linear equations over semigroups and recursively enumerable sets, providing variants of Matiyasevich's Diophantine representation and deriving undecidability results for certain algebraic structures.

Contribution

It introduces new variants of the universal Diophantine representation using linear equations with one unknown, extending previous work to semigroup contexts.

Findings

Variants of Matiyasevich's representation using linear equations

Undecidability results for linear equations over morphism semigroups

Undecidability results for matrix semigroups

Abstract

We study connections between linear equations over various semigroups and recursively enumerable sets of positive integers. We give variants of the universal Diophantine representation of recursively enumerable sets of positive integers established by Matiyasevich. These variants use linear equations with one unkwown instead of polynomial equations with several unknowns. As a corollary we get undecidability results for linear equations over morphism semigoups and over matrix semigroups.