On $τ$-closed $n$-multiply $σ$-local formations of finite groups

Abstract: All groups under consideration are finite. Let $\sigma ={\sigma_i \mid i\in I }$ be some partition of the set of $\mathbb{P}$, $G$ be a group, and $\mathfrak F$ be a class of groups. Then $\sigma (G)={\sigma_i\mid \sigma_i\cap \pi (G)\ne \emptyset} $ and $\sigma (\mathfrak F)=\cup_{G\in \mathfrak F}\sigma (G).$ A function $f$ of the form $f:\sigma \to{\text{formations of groups}}$ is called a formation $\sigma$-function. For any formation $\sigma$-function $f$ the class $LF_\sigma(f)$ is defined as follows: $$ LF_\sigma(f)=(G \text{ is a group } \mid G=1 \text{ or } G\ne 1\ \text{ and }\ G/O_{\sigma_i', \sigma_i}(G) \in f(\sigma_i) \text{ for all } \sigma_i \in \sigma(G)). $$ If for some formation $\sigma$-function $f$ we have $\mathfrak F=LF_\sigma(f),$ then $\mathfrak F$ is called $\sigma$-local, $f$ is called a $\sigma$-local definition of $\mathfrak F.$ Every formation is called 0-multiply $\sigma$-local. For $n > 0,$ a formation $\mathfrak F$ is called $n$-multiply $\sigma$-local provided either $\mathfrak F=(1)$ or $\mathfrak F=LF_\sigma(f),$ where $f(\sigma_i)$ is $(n-1)$-multiply $\sigma$-local for all $\sigma_i\in \sigma(\mathfrak F).$ Let $\tau(G)$ be a set of subgroups of $G$ such that $G\in \tau(G)$. Then $\tau$ is called a subgroup functor if for every epimorphism $\varphi$ : $A \to~B$ and any groups $H\in\tau(A)$ and $T\in\tau(B)$ we have $H^{\varphi}\in\tau(B)$ and $T^{{\varphi}^{-1}}\in\tau(A)$. A class $\mathfrak F$ is called $\tau$-closed if $\tau(G)\subseteq\mathfrak F$ for all $G\in\mathfrak F$. We describe some properties of $\tau$-closed $n$-multiply $\sigma$-local formations, as well as we prove that the set $l^{\tau}_{\sigma_n}$ of all $\tau$-closed $n$-multiply $\sigma$-local formations forms a complete modular algebraic lattice. In addition, we proof that $l^{\tau}_{\sigma_n}$ is $\sigma$-inductive and $\mathfrak G$-separable.