Narrow Results

In this paper we prove that the pseudovariety of Abelian groups is hyperdecidable and moreover that it is completely tame. This is a consequence of the fact that a system of group equations on a free Abelian group with certain rational constraints is solvable if and only if it is solvable in every finite quotient.

In this paper we prove that the pseudovariety LSl of local semilattices is completely κ-reducible, where κ is the implicit signature consisting of the multiplication and the ω-power. Informally speaking, given a finite equation system with rational constraints, the existence of a solution by pseudowords of the system over LSl implies the existence of a solution by κ-words of the system over LSl satisfying the same constraints.