Boolean products of indecomposables. (Q1771965)

It was proved recently by D. Vaggione that discriminator varieties \(V\) are characterized by two properties: (a) the Fraser-Horn property (FHP) and (b) \(V = I\Gamma ^{a}(V_{DI})\), where \(I\) denotes closure under isomorphism, \(\Gamma ^{a}\) denotes closure under formation of Boolean products with clopen equalizers, and \(V _{DI}\) is the class of directly indecomposable members of \(V\). He asked whether FHP could be replaced by the weaker property (BFC) of Boolean Factor Congruences. The author proves that if \(V\) satisfies (BFC) and \(V = I\Gamma ^{a}(V_{DI})\) then \(V_{DI}\) consists of simple algebras and it is a class defined by universal sentences.