О НЕКОТОРЫХ СВОЙСТВАХ РЕШЕТКИ ТОТАЛЬНО σ-ЛОКАЛЬНЫХ ФОРМАЦИЙ КОНЕЧНЫХ ГРУПП

Abstract

Throughout this paper, all groups are finite. Let (Formula Presented) I be some partition of the set of all primes. If n is an integer, G is a group, and F is a class of groups, then (Formula Presented). A function f of the form f: σ → {formations of groups} is called a formation σ-function. For any formation σ-function f the class LFσ f () is defined as follows: (Formula Presented).. If for some formation σ-function f we have (Formula Presented), then the class (Formula Presented) is called σ-local and f is called a σ-local definition of F. Every formaton is called 0-multiply σ-local. For n > 0, a formation F is called n-multiply σ-local provided either F = (1) is the class of all identity groups or (Formula Presented) multiply σ-local for all (Formula Presented). A formation is called totally σ-local if it is n-multiply σ-local for all non-negative integer n. The aim of this paper is to study properties of the lattice of totally σ-local formations. In particular, we prove that the lattice of all totally σ-local formations is algebraic and distributive.