The join \(\mathcal V_1\vee\mathcal V_2\) of two varieties over disjoint languages is the variety axiomatized by the union of the identities satisfied by \(\mathcal V_1\) and \(\mathcal V_2\). The paper provides evidence for the conjecture (after O. C. Garcia and W. Taylor [Mem. Am. Math. Soc. 305 (1984; Zbl 0559.08003)]) that if \(\mathcal V_1\vee\mathcal V_2\) is congruence modular, then one of \(\mathcal V_1\) or \(\mathcal V_2\) already is.
The authors prove the conjecture in the case of idempotent \(\mathcal V_i\) axiomatized by linear identities (i.e., in which terms have at most one function symbol). The main tool for achieving this is the test of congruence modularity appearing in [T. Dent et al., Algebra Univers. 67, No. 4, 375-392 (2012; Zbl 1259.08004)], where an operation of “derivative” on sets of identities \(\Sigma\) is defined such that if an idempotent variety \(\mathcal V\) satisfies some \(\Sigma\) having an inconsistent derivative \(\Sigma'\), then \(\mathcal V\) is congruence modular.
The key step in their proof is to show that for linear idempotent \(\Sigma_i\), one has \((\Sigma_1\cup\Sigma_2)'=\Sigma_1'\cup\Sigma_2'\), and this leads to their result.
The authors extend similarly their theorem to other classes of linear idempotent varieties, namely 1) those satisfying a nontrivial congruence identity; and 2) those \(n\)-permutable for some \(n\).