Every decidable pseudovariety of abelian groups is completely tame

TL;DR

This paper proves that all decidable, proper, non-locally finite pseudovarieties of abelian groups are completely tame when considering an expanded implicit signature, linking decidability directly to tameness.

Contribution

It establishes that every decidable, proper, non-locally finite pseudovariety of abelian groups is completely tame with respect to an enlarged implicit signature, extending previous results.

Findings

Decidable pseudovarieties of abelian groups are completely tame with an enlarged signature

Tameness characterizes decidability in pseudovarieties of abelian groups

The result applies to all proper, non-locally finite pseudovarieties

Abstract

It has been shown that the proper, non-locally finite pseudovarieties of abelian groups are not tame with respect to the canonical signature. In this paper, we show that every decidable, proper, non-locally finite pseudovariety of abelian groups is completely tame with respect to a further enlarged implicit signature. This theorem yields as a corollary that a pseudovariety of abelian groups is decidable if and only if it is completely tame.