Joins and subdirect products of varieties

Graetzer, Lakser and Plonka defined varieties V_1 and V_2 (of the same similarity type) to be independent if there is a binary term x * y such that x * y = x holds in V_1 and x * y = y holds in V_2. They proved that independent subvarieties V_1, V_2 of a variety V are disjoint (i.e., their intersection is the trivial variety) and such that the varietal join of V_1 and V_2 is the direct product V_1 x V_2 (i.e., each algebra in the join decomposes as a direct product of an algebra from V_1 and an algebra from V_2). Jonsson and Tsinakis provided a partial converse to this result: if V is congruence permutable and V_1, V_2 are disjoint, then they are independent (and so V_1 v V_2 = V_1 x V_2). We generalise these results in three somewhat different directions. We show that: (i) If V is subtractive, then Jonsson and Tsinakis' result holds under some minimal assumptions. (ii) If V satisfies some weakened permutability conditions, then disjointness implies a generalised notion of independence and V_1 v V_2 is the subdirect product of V_1 and V_2 (i.e., each algebra in the join decomposes as a subdirect product of an algebra from V_1 and an algebra from V_2). (iii) The same holds if V is congruence 3-permutable. In the process we define a notion of quasi-independence for varieties. Some trivial and not-so-trivial applications of these results will also be presented.