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This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for finite monoids. After rewieving the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups.… (More)

We prove that if π is a recursive set of primes, then pointlike sets are decidable for the pseudovariety of semigroups whose subgroups are π-groups. In particular, when π is the empty set, we obtain Henck-ell's decidability of aperiodic pointlikes. Our proof, restricted to the case of aperiodic semigroups, is simpler than the original proof.

We give a short proof, using profinite techniques, that idem-potent pointlikes, stable pairs and triples are decidable for the pseu-dovariety of aperiodic monoids. Stable pairs are also described for the pseudovariety of all finite monoids.

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