Linear equations and recursively enumerable sets

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.