Undecidable problems for completely 0-simple semigroups

The decidability of the uniform word problem in a pseudovariety V of groups is known to be equivalent to the decidability of a number of embedding problems for finite semigroups in classes related to completely 0-simple semigroups with subgroups from V. Unfortunately the existing proof of some of these equivalences turns out to contain an error. We provide an update on these types of properties, correcting the error and amending results accordingly. Several new undecidability properties are exhibited, some superceding previous results. A theme of many of the new contributions is problems that are solvable in polynomial time but become undecidable when a facet of their description is restricted by some finiteness condition. © 2009 Elsevier B.V