Eventually image-Related Sets and Systems of Equations over Finite Semigroups and Rings Original Research Article

I prove that given a finite semigroup or finite associative ringSand a system Σ of equations of the formax=borxa=b, wherea, bset membership, variantS,xis an unknown, it is algorithmically impossible to decide whether or not Σ is solvable overS, that is, whether or not there exists a bigger semigroup or ring (resp. finite semigroup, finite ring)T>Ssuch that Σ has a solution inT. The proof employs the unsolvability of the uniform word problem in the case of groups (Novikov) and in the class of finite groups (Slobodskoii) and the so-called split systems.