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

All groups under consideration are finite. Let σ = { σ i | i ∈ I } be some partition of the set of all primes P , G be a group, F be a class of groups, σ (G) = { σ i | σ i ∩ π (G) ≠ ∅ } , and σ (F) = ∪ G ∈ F σ (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 L F σ (f) is defined as follows: L F σ (f) = (G is a group |G = 1 or G ≠ 1 and G/O σ i ′ , σ i (G) ∈ f (σ i) for all σ i ∈ σ (G)). If for some formation σ-function f we have F = L F σ (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 = L F σ (f) , where f (σ i) is (n − 1) -multiply σ-local for all σ i ∈ 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 H φ ∈ τ (B) and T φ − 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 (x 1 , ... , x m) signatures { ∩ , ∨ θ } , any θ-formations F 1 , ... , F m and any group A ∈ ν (F 1 , ... , F m) there are groups A 1 ∈ F 1 , ... , A m ∈ F m such that A ∈ ν (θ form A 1 , ... , θ form A m) . We prove that the lattice of all τ-closed n-multiply σ-local formations is a separable lattice of formations. [ABSTRACT FROM AUTHOR]