POINTLIKE SETS, HYPERDECIDABILITY AND THE IDENTITY PROBLEM FOR FINITE SEMIGROUPS

Abstract

In this paper, we give a relationship between the identity problem and the problem of deciding whether certain subsets of nilpotent semigroups are pointlike. We then use this to give an example of a pseudovariety which has a decidable membership problem, but for which one cannot decide pointlike sets. Then, by modifying the equations, we show that no graph is fundamentally hyperdecidable by constructing, for each graph, a labeling over a nilpotent semigroup for which we cannot decide inevitability with respect to the pseudovariety defined by these equations.