On some properties of the lattice of totally $\sigma$-local formations of finite groups

Throughout this paper, all groups are finite. Let $\sigma=\{\sigma_{i}|i\in I\}$ be some partition of the set of all primes $\mathbb{P}$. If $n$ is an integer, $G$ is a group, and $\mathfrak{F}$ is a class of groups, then $\sigma(n)=\{\sigma_{i}|\sigma_{i} \cap \pi(n)\neq \varnothing\}$, $\sigma(G)=\sigma(|G|)$ and $\sigma(\mathfrak{F})=\cup_{G\in \mathfrak{F}} \sigma(G)$. A function $f$ of the form $f:\sigma\rightarrow$ {formations of groups} is called a formation $\sigma$-function. For any formation $\sigma$-function $f$ the class $LF_{\sigma}(f)$ is defined as follows:
$LF_{\sigma}(f)=(G|G=1$ или $G\neq 1$ и $ G\backslash O_{\sigma'_{i},\sigma_{i}}(G)\in f(\sigma_{i})$ для всех $\sigma_{i}\in \sigma(G))$.
If for some formation $\sigma$-function $f$ we have $\mathfrak{F}=LF_{\sigma}(f)$, then the class $\mathfrak{F}$ is called $\sigma$-local and $f$ is called a $\sigma$-local definition of $\mathfrak{F}$. Every formaton is called $0$-multiply $\sigma$-local. For $n > 0$, a formation $\mathfrak{F}$ is called $n$-multiply $\sigma$-local provided either $\mathfrak{F} = (1)$ is the class of all identity groups or $\mathfrak{F} = LF_{\sigma}(f)$, where $f(\sigma_{i})$ is $(n-1)$-multiply $\sigma$-local for all $\sigma_{i}\in \sigma(\mathfrak{F})$. A formation is called totally $\sigma$-local if it is $n$-multiply $\sigma$-local for all non-negative integer $n$. The aim of this paper is to study properties of the lattice of totally $\sigma$-local formations. In particular, we prove that the lattice of all totally$\sigma$-local formations is algebraic and distributive.