On some properties of the lattice of partially totally saturated formations of finite groups

All groups under consideration are finite. The paper studies some properties of the lattice of all $\tau$ -closed totally $\omega$ -saturated formations. We show that for any subgroup functor $\tau$, the lattice of all $\tau$ -closed totally $\omega$ -saturated formations is modular and algebraic. We also prove that the lattice of all totally $\omega$ -saturated formations is G -separable. This strengthens a theorem of V.G. Safonov. Using embeddability the lattice of all $\tau$ -closed totally $\omega$ -saturated formations in the lattice of all totally $\omega$ -saturated formation, we establish that the lattice of all $\tau$ -closed totally $\omega$ -saturated formations is $G-$separable. In particular, we show that the lattice of all $\tau$ -closed totally $\rho$ -saturated formations is modular, algebraic, and $G-$separable as well as the lattice of all $\tau$ -closed totally saturated formations.