On independent varieties and some related notions (Q382342)

Two subvarieties \(V_1, V_2\) of a given variety \(V\) are \textit{disjoint} if their intersection consists of trivial algebras. They are \textit{independent} if there is a binary term \(x \ast y\) such that \(V_1\) satisfies \(x \ast y = x\) and \(V_2\) satisfies \(x \ast y = y\). The \textit{direct product} \(V_1 \times V_2\) of \(V_1\) and \(V_2\) is the class \(\{A_1 \times A_2 \mid A_1 \in V_1, A_2 \in V_2\}\), and the subdirect product \(V_1 \times_s V_2\) is the class of subdirect products \(A_1 \times_s A_2\) of some \(A_1 \in V_1\), \(A_2 \in V_2\). It is well known that if \(V_1\) and \(V_2\) are independent, then they are disjoint and \(V_1 \vee V_2 = V_1 \times V_2\). Also, if \(V\) is a congruence permutable variety, and \(V_1, V_2\) are disjoint, then \(V_1 \vee V_2 = V_1 \times V_2\). The authors provide a procedure for constructing an independent companion of a given solvable subvariety of a congruence modular variety, they show that joins of independent varieties inherit Mal'tsev properties from their joinands and investigate independence in 3- and 4-permutable varieties. The concept of independent varieties \(V_1\) and \(V_2\) is then generalized. The varieties \(V_1\) and \(V_2\) are called \textit{quasi-independent} if there are unary so-called \(V_1 \vee V_2\)-compatible and \(V_1 \vee V_2\)-idempotent terms \(t_1(x)\) and \(t_2(x)\), and a binary term \(x \ast y\), such that \(V_1\) satisfies \(x \ast y = t_1(x)\), \(t_2(x) = x\) and \(V_2\) satisfies \(x \ast y = t_2(y)\), \(t_1(x) = x\). Quasi-independent varieties \(V_1\) and \(V_2\) satisfying certain additional quasi-equations are called \textit{orthogonal}, and quasi-independent and orthogonal varieties with some more additional conditions are called \textit{strictly quasi-independent}. The authors show that if subvarieties \(V_1\) and \(V_2\) of a variety \(V\) are strictly quasi-independent, then \(V_1 \vee V_2 = V_1 \times_s V_2\). They also show that joins of strictly quasi-independent varieties inherit the congruence extension property and the strong amalgamation property from the joinands, and conversely. The last section provides a number of application, in particular for Boolean algebras with operators and for groups.