Joins and subdirect products of varieties (Q634765)

According to a classical theorem by \textit{G. Grätzer, H. Lakser} and \textit{J. Płonka} [``Joins and direct products of equational classes'', Can. Math. Bull. 12, 741--744 (1969; Zbl 0188.04903)], independent varieties have two rather pleasant properties: (i) they are disjoint, and (ii) \(\mathcal{V}_{1}\vee \mathcal{V}_{2}=\mathcal{V}_{1}\times \mathcal{V}_{2}\), i.e., every algebra from the join \(\mathcal{V}_{1}\vee \mathcal{V}_{2}\) directly decomposes into members of \(\mathcal{V}_{1}\) and \( \mathcal{V}_{2}\). In a more recent paper [``Products of classes of residuated structures'', Stud. Log. 77, No. 2, 267--292 (2004; Zbl 1072.06003)], \textit{B. Jónsson} and \textit{C. Tsinakis} proved a partial converse of this result, which holds for congruence-permutable varieties: If \(\mathcal{V}\) is a congruence-permutable variety and \(\mathcal{V}_{1}\), \( \mathcal{V}_{2}\) are disjoint subvarieties of it, then they are independent. Taken together, the above results imply the following: If \(\mathcal{V}\) is a congruence-permutable variety and \(\mathcal{V}_{1}\), \( \mathcal{V}_{2}\) are disjoint subvarieties of it, then \(\mathcal{V}_{1}\vee \mathcal{V}_{2}=\mathcal{V}_{1}\times \mathcal{V}_{2}\). Here, this last result is generalized in three different directions. First, the authors observe that if \(\mathcal{V}\) is pointed, then the congruence-permutability hypothesis in this result is too strong: under some quite minimal assumption, it suffices that \(\mathcal{V}\) be subtractive. Then, they lift the previous restriction on the type \(\nu\), which now need not contain any constant, and show that under appropriate weak permutability conditions any two disjoint varieties \(\mathcal{V}_{1}\), \( \mathcal{V}_{2}\) fall under a generalized relation of quasi-independence. Finally, they prove that the same holds if \(\mathcal{V}\) is congruence-3-permutable.