Every decidable pseudovariety of abelian groups is completely tame (Q2003184)

Let \(\mathbf V\) be a pseudovariety of finite semigroups. \(\mathbf V\) is said to be decidable if there is an algorithm to test for every finite semigroup \(S\) the membership \(S\in\mathbf V\). Since the decidability is not preserved by some operations such as semidirect product, join and Mal'cev product of pseudovarieties, a notion of tameness has been introduced by \textit{J. Almeida} and \textit{B. Steinberg} as a ``refined version of decidability'' [Proc. Lond. Math. Soc. (3) 80, No. 1, 50--74 (2000; Zbl 1027.20033)]. Using tameness of pseudovarieties it is possible to establish the decidability of pseudovarieties obtained by application of the mentioned operations. It is shown that ``every decidable, proper, non-locally finite pseudovariety of abelian groups is completely tame with respect to a further enlarged implicit signature''. It is worth mentioning the following corollary. ``Every pseudovariety of finite abelian groups is decidable if and only if it is completely tame.''