On separability of the lattice of τ -closed n -multiply σ -local formations

All groups under consideration are finite. Let σ={σi|i∈I} be some partition of the set of all primes P, G be a group, F be a class of groups, σ(G)={σi|σi∩π(G)≠∅}, and σ(F)=∪G∈Fσ(G).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: LFσ(f)=(G is a group |G = 1 or G≠1 and G/Oσi′,σi(G)∈f(σi) for allσi∈σ(G)). If for some formation σ-function f we have F=LFσ(f), then the class F is called σ-local and f is called a σ-local definition of F. Every formation 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 F=LFσ(f), where f(σi) is (n−1)-multiply σ-local for all σi∈σ(F). Let τ(G) be a set of subgroups of G such that G∈τ(G). Then τ is called a subgroup functor if for every epimorphism φ: A→B and any groups H∈τ(A) and T∈τ(B) we have Hφ∈τ(B) and Tφ−1∈τ(A). A formation of groups F is called τ-closed if τ(G)⊆F for all G∈F. A complete lattice of formations θ is called separable, if for any term ν(x1,…,xm) signatures {∩,∨θ}, any θ-formations F1,…,Fm and any group A∈ν(F1,…,Fm) there are groups A1∈F1,…,Am∈Fm such that A∈ν(θform A1,…,θform Am). We prove that the lattice of all τ-closed n-multiply σ-local formations is a separable lattice of formations.