Smith introduced a notion of commutators for congruence permutable (that is, Malcev) varieties, developed in the monograph "Mal'cev Varieties". This notion was generalised by Hagemann and Herrmann in "A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity", taking in consideration the more general class of congruence-modular varieties. Later, Freeze and McKenzie presented in "Commutator Theory for Congruence Modular Varieties" a different point of view on the notion of commutators, but showing that their coincide with the one introduced by Hagemann and Herrmann.
- Do the notion introduced by Freeze and McKenzie (and so the one of Hagemann and Herrmann) coincide with the one of Smith for congruence permutable varieties? While this seems reasonable, I have some troubles in understanding the technicalities (and the notation) of the description from Smith to compare the two, and I have not find a clear sentence or result asserting this.
- Suppose that moreover our variety is also a variety of $\Omega$-group (in the sense of Higgins), so that we have at our disposal also Higgins commutator ("Groups with Multiple Operators"). Any chances in general that Smith and Higgins coincide? There are some results in this directions?