Separability of the lattice of τ-closed totally ω-composition formations of finite groups

Abstract

Let $\mathfrak{X}$ be a non-empty class of finite groups. A complete lattice $\theta$ of formations is said $\mathfrak{X}$-separable if for every term $\eta(x_1, \ldots , x_n)$ of signature $\{\cap, \vee_{\theta}\}$, $\theta$-formations $\mathfrak{F}_1, \ldots , \mathfrak{F}_n$, and every group $A\in \mathfrak{X}\cap \eta(\mathfrak{F}_1, \ldots , \mathfrak{F}_n)$ are exists $\mathfrak{X}$-groups $A_1\in\mathfrak{F}_1, \ldots , A_n\in\mathfrak{F}_n$ such that $A\in\eta(\theta\textup{form}(A_1), \ldots , \theta\textup{form}(A_n))$. In particular, if $\mathfrak{X}=\mathfrak{G}$ is the class of all finite groups then the lattice $\theta$ of formations is said $\mathfrak{G}$-separable or, briefly, separable. It is proved that the lattice $c^{\tau}_{\omega_\infty}$ of all $\tau$-closed totally $\omega$-composition formations is $\mathfrak{G}$-separable.