ON THE DECIDABILITY OF MEMBERSHIP IN THE GLOBAL OF A MONOID PSEUDOVARIETY
Karl Auinger · International Journal of Algebra and Computation · 2010
It is proved that the global operator g : V ↦ gV does not preserve decidability of membership. More specifically, some of Kad'ourek's results on the locality of the pseudovariety DG in combination with a construction by Steinberg and the author are used to find a pseudovariety H of groups, with decidable membership, for which membership in the category pseudovariety gDH is undecidable. This answers a question raised by Rhodes and Steinberg. As a consequence, the operator V ↦ V * D = V * LI does not preserve decidability of membership either — this refutes an old conjecture by Straubing.