Weak containment and maximal sofic approximations

We show that the class of sofic actions is closed under direct products and contains a (non-unique) maximal element in the weak containment order. For any sofic group we construct nice sofic approximations such that all the sofic actions are approximable by them in a doubly-quenched way. We use recent result by Hayes to establish for these sofic approximations the equality of sofic measure entropy to the topological one for algebraic actions whenever the former is not $-\infty$. We also use his another result to establish the product formula for Pinsker factors of these approximations.