On τ-closed n-multiply ω-composition formations with Boolean sublattices

In the universe of finite groups the description of τ-closed n-multiply ω-composition formations with Boolean sublattices of τ-closed n-multiply ω-composition subformations is obtained.

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Дата:	2010
Автор:	Zhiznevsky, P.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2010
Назва видання:	Algebra and Discrete Mathematics
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Цитувати:	On τ-closed n-multiply ω-composition formations with Boolean sublattices / P. Zhiznevsky // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 118–127. — Бібліогр.: 11 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1548722019-06-17T01:31:07Z On τ-closed n-multiply ω-composition formations with Boolean sublattices Zhiznevsky, P. In the universe of finite groups the description of τ-closed n-multiply ω-composition formations with Boolean sublattices of τ-closed n-multiply ω-composition subformations is obtained. 2010 Article On τ-closed n-multiply ω-composition formations with Boolean sublattices / P. Zhiznevsky // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 118–127. — Бібліогр.: 11 назв. — англ. 2000 Mathematics Subject Classification:20D10, 20F17. http://dspace.nbuv.gov.ua/handle/123456789/154872 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In the universe of finite groups the description of τ-closed n-multiply ω-composition formations with Boolean sublattices of τ-closed n-multiply ω-composition subformations is obtained.
format	Article
author	Zhiznevsky, P.
spellingShingle	Zhiznevsky, P. On τ-closed n-multiply ω-composition formations with Boolean sublattices Algebra and Discrete Mathematics
author_facet	Zhiznevsky, P.
author_sort	Zhiznevsky, P.
title	On τ-closed n-multiply ω-composition formations with Boolean sublattices
title_short	On τ-closed n-multiply ω-composition formations with Boolean sublattices
title_full	On τ-closed n-multiply ω-composition formations with Boolean sublattices
title_fullStr	On τ-closed n-multiply ω-composition formations with Boolean sublattices
title_full_unstemmed	On τ-closed n-multiply ω-composition formations with Boolean sublattices
title_sort	on τ-closed n-multiply ω-composition formations with boolean sublattices
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/154872
citation_txt	On τ-closed n-multiply ω-composition formations with Boolean sublattices / P. Zhiznevsky // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 118–127. — Бібліогр.: 11 назв. — англ.
series	Algebra and Discrete Mathematics
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first_indexed	2025-07-14T06:56:23Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 10 (2010). Number 2. pp. 118 – 127 c© Journal “Algebra and Discrete Mathematics” On τ -closed n-multiply ω-composition formations with Boolean sublattices Pavel Zhiznevsky Communicated by L. A. Shemetkov Abstract. In the universe of finite groups the descrip- tion of τ -closed n-multiply ω-composition formations with Boolean sublattices of τ -closed n-multiply ω-composition subformations is obtained. Introduction Let H be a nonempty nilpotent saturated formation, and let F be a nonempty τ -closed n-multiply ω-composition formation such that F 6⊆ H. Then F/τωn F ∩ H denotes the sublattice of all τ -closed n-multiply ω- composition formations M such that F ∩ H ⊆ M ⊆ F. In this paper we obtain the description of τ -closed n-multiply ω-composition formations F such that F 6⊆ H and the lattice F/τωn F ∩ H is Boolean, where H is a nonempty nilpotent saturated formation. This is the solution of the following problem [1]: Problem (A.N. Skiba, L.A. Shemetkov). Describe τ -closed n- multiply ω-composition formations F 6⊆ H such that the lattice F/τωn F∩H is Boolean, where H is a nonempty nilpotent saturated formation. The solution of that problem in the case, when τ is the trivial subgroup functor (i.e., τ(G) = {G} for all groups G) and H = N is the formation of all nilpotent groups is obtained in paper [2]. In the other case, when τ is the trivial subgroup functor, n = 1 and H = N, the solution of that problem was given in [3]. 2000 Mathematics Subject Classification: 20D10, 20F17. Key words and phrases: finite group, formation, τ -closed n-multiply ω- composition formation, Boolean lattice, complemented lattice. P. A. Zhiznevsky 119 1. Preliminaries All groups considered in this paper are finite. We use the terminology of [1, 4, 5, 6]. Here, we recall some definitions and notation. Let ω be a nonempty set of prime numbers. Every function of the form f : ω ∪ {ω′} → {formations of groups} is called an ω-composition satellite. Following Doerk and Hawkes [7] we use Cp(G) to denote the intersection of all centralizers of abelian chief p-factors of G (we note that Cp(G) = G if G has no such chief factors). If X is a set of groups, then we use Com+(X) to denote the class of all abelian simple groups A such that A ≃ H/K for some composition factor H/K of a group G ∈ X. We write Com+(G) for the set Com+({G}). Let f be an ω-composition satellite. Then following [1] we put CFω(f) = { G \| G/Rω(G) ∈ f(ω′) and G/Cp(G) ∈ f(p) for all p ∈ π(Com(G)) ∩ ω} . Here Rω(G) denotes the largest normal soluble ω-subgroup of G. If F is a formation such that F = CFω(f) for some ω-composition satellite f , then F is called an ω-composition formation, and f is its an ω-composition satellite. Every group formation is considered as 0-multiply ω-composition formation. For n ≥ 1, a formation F is called n-multiply ω-composition formation, if F = CFω(f), where all values of an ω-composition satellite f are (n− 1)-multiply ω-composition formations. Let τ be a functor such that for any group G, τ(G) is a set of subgroups of G, and G ∈ τ(G). Following [5] we say that τ is 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 group F is called τ -closed if τ(G) ⊆ F for all G ∈ F. The set of all τ -closed n-multiply ω-composition formations is denoted by cτωn . A satellite f is called a cτωn−1 -valued ω-composition sattelite if all values of f belong to cτωn−1 . A τ -closed n-multiply ω-composition formation F is called Hτωn -criti- cal (or a minimal τ -closed n-multiply ω-composition non-H-formation) if F * H but all proper τ -closed n-multiply ω-composition subformations of F are contaned in H. Let X be a set of groups. Then cτωn formX is the τ -closed n-multiply ω- composition formation generated by X, i.e., cτωn formX is the intersection of all τ -closed n-multiply ω-composition formation containing X. If X = {G}, then cτωn formX = cτωn formG is called an one-generated τ -closed n-multiply ω-composition formation. 120 On cτωn -formations with Boolean sublattices For any set {Fi \| i ∈ I} of τ -closed n-multiply ω-composition forma- tions, we put ∨τωn (Fi \| i ∈ I) = cτωn form( ⋃ i∈I Fi). In particulary, M ∨τωn H = cτωn form (M ∪ H). A lattice is called modular if and only if its elements satisfy the modular identity: if x ≤ z, then x ∨ (y ∧ z) = (x ∨ y) ∧ z. A lattice is called distributive if and only if it satisfies the following: L1. (x ∧ y) ∨ (y ∧ z) ∨ (z ∧ x) = (x ∨ y) ∧ (y ∧ z) ∧ (z ∧ x), L2. x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), L3. x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). An element which covers 0 in a partly ordered set P (i.e., a minimal element in the subset of P obtained by excluding 0) is called an atom. By a complement of an element x of a lattice L with 0 and 1 is meant an element y ∈ L such that x∧y = 0 and x∨y = 1; L is called complemented if all its elements have complements. In the paper we consider only subgroup functors τ such that the set τ(G) is contained in the set of all subnormal subgroups of G, for any group G. Lemma 1 ([8], Theorem 2). The lattice cτωn of all τ -closed n-multiply ω-composition formations is modular for any non-negative number n. Lemma 2 ([6, p. 65]). Any sublattice of a modular lattice is modular. Lemma 3 ([6, p. 73], Theorem 6). Let L be any modular lattice, and let u and v be any two elements of L. Then the correspondences x→ u ∨ x and y → v ∧ y are inverse isomorphisms between [u ∧ v, v] and [u, u ∨ v]. Moreover, they carry quotients in these intervals into transposed quotients. Lemma 4 ([9], Theorem 2). Let H be a nonempty nilpotent saturated formation, and let F be a τ -closed n-multiply ω-composition formation such that F 6⊆ H (n ≥ 1). Then Hτωn -defect of the formation F is equal 1 if and only if F = M ∨τωn K, where M ⊆ H, K is a minimal τ -closed n-multiply ω-composition non-H-formation, besides: 1) if τ -closed n-multiply ω-composition subformation F1 of F, such that F1 ⊆ H, then F1 ⊆ M ∨τωn (K ∩ H); 2) if τ -closed n-multiply ω-composition subformation F1 of F, such that F1 6⊆ H, then F1 = K ∨τωn (F1 ∩ H). P. A. Zhiznevsky 121 Lemma 5 ([9], Lemma 17). Let H be a nonempty nilpotent saturated formation, M ⊆ H, and let Ω = {Ki \| i ∈ I} be some set of minimal τ - closed n-multiply ω-composition non-H-formations. If K is some τ -closed n-multiply ω-composition non-H-formation of M∨τωn (∨τωn Ki \| i ∈ I), then K ∈ Ω. Lemma 6 ([9], Theorem 4). Let H be a nonempty nilpotent saturated formation, and let F be an one-generated τ -closed n-multiply ω-composi- tion formation such that F 6⊆ H (n ≥ 1). If F/τωn F ∩ H is a complemented lattice and M is an element of F/τωn F ∩ H, then M = ∨τωn ((F ∩ H) ∨τωn Ki \| i ∈ I), where {Ki \| i ∈ I} is the set of all minimal τ -closed n-multiply ω-composi- tion non-H-subformations of M. Lemma 7 ([9], Theorem 1). Let H be a nonempty nilpotent saturated formation, and let F be a τ -closed n-multiply ω-composition formation such that F 6⊆ H (n ≥ 1). Then F has at least one a minimal τ -closed n-multiply ω-composition non-H-subformation. Lemma 8 ([9], Lemma 24). Let H be a nonempty nilpotent saturated formation, and let F be a τ -closed n-multiply ω-composition formation such that F 6⊆ H (n ≥ 1). Then M is an atom of the lattice F/τωn F ∩ H if and only if M = (F ∩H) ∨τωn K, where K is a minimal τ -closed n-multiply ω-composition non-H-subformation of F. Lemma 9 ([10, p. 50], Lemma 9). The following inequalities hold in any lattice: 1) (x ∧ y) ∨ (x ∧ z) ≤ x ∧ (y ∨ z); 2) x ∨ (y ∧ z) ≤ (x ∨ y) ∧ (x ∨ z); 3) (x ∧ y) ∨ (y ∧ z) ∨ (z ∧ x) ≤ (x ∨ y) ∧ (y ∨ z) ∧ (z ∨ x); 4) (x ∧ y) ∨ (x ∧ z) ≤ x ∧ (y ∨ (x ∧ z)). The first three are called distributive inequalities, and the last is the modular inequality. 2. The Main result Lemma 10. Let F ∩ H ⊆ M1 ⊆ M ⊆ F, where M1, M, F are τ -closed n-multiply ω-composition formations, and let H be a nonempty nilpotent saturated formation (n ≥ 1). If H1 is a complement of M1 in the lattice F/τωn F∩H, then M∩H1 is a complement of M1 in the lattice M/τωn M∩H. 122 On cτωn -formations with Boolean sublattices Proof. By the conditions of the lemma we have that M1∩H1 = F∩H and M1 ∨ τ ωn H = F. It follows from Lemmas 1 and 2 that M = M ∩ (M1 ∨ τ ωn H1) = M1∨ τ ωn (M∩H1). Moreover, it is clear that (M∩H1)∩M1 = M∩H. Hence M ∩ H1 is a complement of M1 in the lattice M/τωn M ∩ H. Lemma 11. Let H be a nonempty nilpotent saturated formation, let F and M be τ -closed n-multiply ω-composition formations such that M ⊆ F (n ≥ 1). If F/τωn F∩H is a complemented lattice, then M/τωn M∩H is also a complemented lattice. Proof. From Lemmas 1 and 2 we see that the lattice F/τωn F∩H is modular. By Lemma 3 we have the following isomorphism of lattices: ((F ∩ H) ∨τωn M)/τωn F ∩ H ≃ M/τωn (M ∩ F ∩ H) = M/τωn M ∩ H. Since ((F ∩ H) ∨τωn M)/τωn F ∩ H is a sublattice of F/τωn F ∩ H, then by Lemma 10, it follows that (F ∩ H ∨τωn M)/τωn F ∩ H is a complemented lattice. Consequently, M/τωn M ∩ H is also a complemented lattice. Lemma 12. Let H be a nonempty nilpotent saturated formation, and let F be a τ -closed n-multiply ω-composition formation such that F 6⊆ H (n ≥ 1). Denote by Ω the set of all τ -closed n-multiply ω-composition formations Fi of F (i ∈ I) such that Fi 6⊆ H and F ∩ H is a maximal τ -closed n-multiply ω-composition subformation of Fi. Put R = cτωn form ( ⋃ i∈I Fi), where Fi ∈ Ω. If M is a τ -closed n-multiply ω-composition subformation of R with maximal subformation F ∩ H and M 6⊆ H, then M ∈ Ω. Proof. Following Lemma 4, for every i ∈ I we have Fi = (F ∩ H) ∨τωn Ki, where Ki is a minimal τ -closed n-multiply ω-composition non-H-formation. Then R = cτωn form ( ⋃ i∈I Fi) = cτωn form ( ⋃ i∈I ((F ∩ H) ∨τωn Ki)) = = cτωn form ((F ∩ H) ∪ (∨τωn Ki \| i ∈ I)) = (F ∩ H) ∨τωn (∨τωn Ki \| i ∈ I). From Lemma 4 we have M = (F ∩ H) ∨τωn K, where K is a minimal τ -closed n-multiply ω-composition non-H-formation. Consequently, by Lemma 5 we get K ∈ {Ki \| i ∈ I}, i.e. K = Ki for some i ∈ I. Thus, M = (F ∩ H) ∨τωn K ∈ {(F ∩ H) ∨τωn Ki \| i ∈ I} = Ω. P. A. Zhiznevsky 123 Theorem 1. Let H be a nonempty nilpotent saturated formation, and let F be a τ -closed n-multiply ω-composition formation such that F 6⊆ H (n ≥ 1). Then the following statements are equivalent: 1) F/τωn F ∩ H is a complemented lattice; 2) F = (F ∩ H) ∨τωn (∨τωn Ki \| i ∈ I), where {Ki \| i ∈ I} is the set of all minimal τ -closed n-multiply ω-composition non-H-subformation of F; 3) F/τωn F ∩ H is a Boolean lattice. Proof. 1)⇒2). Let M be an one-generated τ -closed n-multiply ω-compo- sition subformation of F. Then by Lemma 11 the lattice M/τωn M ∩ H is complemented. From Lemma 6 we have: M = ∨τωn ((M ∩ H) ∨τωn Ki \| i ∈ I) = (M ∩ H) ∨τωn (∨τωn Kj \| j ∈ J), where {Kj \| j ∈ J ⊆ I} is the set of all minimal τ -closed n-multiply ω- composition non-H-subformations ofM. Obviously, any τ -closed n-multiply ω-composition formation is a join (in the lattice cτωn ) of all proper one- generated τ -closed n-multiply ω-composition subformations, i.e., F = ∨τωn (cτωn formG \| G ∈ F) = ∨τωn (Mi \| i ∈ I). Then F = ∨τωn (Mi \| i ∈ I) = ∨τωn ((Mi ∩ H) ∨τωn (∨τωn Kj \| j ∈ Ji) \| i ∈ I) = = ∨τωn (Mi ∩ H \| i ∈ I) ∨τωn (∨τωn Ki \| i ∈ I). For every i ∈ I we have Mi ⊆ F. Then Mi ∩ H ⊆ F ∩ H. Therefore, ∨τωn (Mi ∩ H \| i ∈ I) ⊆ F ∩ H. Suppose that F∩H 6⊆ ∨τωn ((Mi)∩H \| i ∈ I), and let G ∈ (F ∩ H) \ ∨τωn (Mi ∩ H \| i ∈ I). Since cτωn formG = (cτωn formG) ∩ H ⊆ ∨τωn (Mi ∩ H \| i ∈ I), we have G ∈ ∨τωn (Mi ∩ H \| i ∈ I). A contradiction. Hence, we get F ∩ H ⊆ ∨τωn (Mi ∩ H \| i ∈ I) and F ∩ H = ∨τωn (Mi ∩ H \| i ∈ I). Thus, F = (F ∩ H) ∨τωn (∨τωn Ki \| i ∈ I), i.e., 2) is true. 2)⇒1). First we shall show that every element of F/τωn F ∩ H is a join of atoms which are contained in that element. Let M be an element of F/τωn F ∩ H. Then M is τ -closed n-multiply ω-composition formation. Let ψ = {Ki \| i ∈ I} be the set of all minimal τ -closed n-multiply ω-composition non-H-subformations of F, ψ1 = {Ki \| i ∈ I1} be a set of all minimal τ -closed n-multiply ω-composition non-H- subformations of F such that Ki ⊆ M for all i ∈ I1 ⊆ I, and let ψ2 is a 124 On cτωn -formations with Boolean sublattices complement to ψ1 in ψ. Then Ri = cτωn form (ψi) is the τ -closed n-multiply ω-composition formation generated by ψi, where i = 1, 2. Since the lattice cτωn is modular, we have: M = M ∩ F = M ∩ ((F ∩ H) ∨τωn (∨τωn Ki \| i ∈ I)) = = (F ∩ H) ∨τωn (M ∩ (∨τωn Ki \| i ∈ I)) = = (F ∩ H) ∨τωn (M ∩ (R1 ∨ τ ωn R2)) = (F ∩ H) ∨τωn R1 ∨ τ ωn (M ∩R2). Assume that M ∩ R2 6⊆ F ∩ H. Then by Lemma 7 M ∩ R2 has a minimal τ -closed n-multiply ω-composition non-H-formation Ki for some i ∈ I. Hence, by Lemma 5 we get Ki ∈ ψ1 ∩ ψ2 = ∅. A contradiction. Therefore, M ∩R2 ⊆ F ∩ H. Thus, M = (F ∩ H) ∨τωn R1 = (F ∩ H) ∨τωn (∨τωn Ki \| Ki ∈ ψ1) = = ∨τωn ((F ∩ H) ∨τωn Ki \| Ki ∈ ψ1). From Lemma 6 we see that every element M of the lattice F/τωn F∩H is a join of an atoms which are contained in M. Now we shall show that every element M of the lattice F/τωn F∩H has a complement. If M = F, then F ∩ H is complement to M in F/τωn F ∩ H. Therefore, we can assume that M 6= F. Denote by Σ the set of all atoms of F/τωn F∩H, and denote by Ω1 a set of all atoms of F/τωn F∩H contained in M. If Σ = Ω1, then M = (F ∩ H) ∨τωn (∨τωn Ki \| i ∈ I) = F. A contradiction. Therefore, Σ 6= Ω1. Let Ω2 be a complement of Ω1 in Σ, and let K = cτωn form (Ω2). We prove that K is a complement of M in F/τωn F ∩ H. Since by the condition F = (F ∩ H) ∨τωn (∨τωn Ki \| i ∈ I), then by Lemma 8 we have F = M ∨τωn K. Let R = M ∩ K. Since M and K are elements of the lattice F/τωn F ∩ H, we have F ∩ H ⊆ R. Assume that R 6⊆ F ∩ H. According to Lemma 7, the formation R has a minimal τ -closed n-multiply ω-composition non-H-formation Ki for some i ∈ I. Hence, (F ∩ H) ∨τωn Ki ⊆ R. By Lemma 4, (F ∩ H) ∨τωn Ki has a maximal τ -closed n-multiply ω-composition subformation which is contained in H. Now by Lemma 12 and applying the result of previous paragraph, we have (F ∩ H) ∨τωn Ki ∈ Ω1 ∩ Ω2 = ∅. A contradiction. Hence, M ∩ K = F ∩ H. Thus, K is a complement of M in F/τωn F ∩H. So, the lattice F/τωn F ∩H is complemented. 3)⇒1). This case is obvious. 1)⇒3). We need to show that for any elements M1, M2, M3 of the lattice F/τωn F ∩ H the following equality is true: M1 ∩ (M2 ∨ τ ωn M3) = (M1 ∩M2) ∨ τ ωn (M1 ∩M3). (∗) P. A. Zhiznevsky 125 According to Lemma 9, for the lattice F/τωn F∩H the following inclusion holds: (M1 ∩M2) ∨ τ ωn (M1 ∩M3) ⊆ M1 ∩ (M2 ∨ τ ωn M3). Put X = M1 ∩ (M2 ∨ τ ωn M3) and Y = (M1 ∩M2)∨ τ ωn (M1 ∩M3). We show X ⊆ Y. By Lemma 11 for every element M of F/τωn F ∩ H a lattice M/τωn M ∩ H is complemented. Since 1)⇒2) is true, it follows that M = (M ∩ H) ∨τωn (∨τωn Ki \| i ∈ I), (∗∗) where {Ki \| i ∈ I} is the set of all minimal τ -closed n-multiply ω-composi- tion non-H-subformations of M. Since X and Y are elements of F/τωn F∩H, (∗∗) is true. Therefore, now we need to show that every minimal τ -closed n-multiply ω-composition non-H-subformation K of X is contained in Y. Denote by ψj the set of all minimal τ -closed n-multiply ω-composition non-H-subformations of Mj , where j = 1, 2, 3. Clearly, K ⊆ M1. From (∗∗) for M1 we have K ∈ ψ1. Besides, we have K ⊆ ( (M2 ∩ H) ∨τωn ( cτωn form ( ⋃ Ki∈ψ2 Ki ))) ∨τωn ∨τωn ( (M3 ∩ H) ∨τωn ( cτωn form ( ⋃ Ki∈ψ3 Ki ))) = = ((M2 ∨ τ ωn M3) ∩ H) ∨τωn ( cτωn form ( ⋃ Ki∈ψ2∪ψ3 Ki )) . From Lemma 12 we get K ∈ ψ2 ∪ ψ3. Then either K ⊆ M1 ∩M2 or K ⊆ M1 ∩M3. Therefore, K ⊆ Y, i.e., equality (∗) is true. So, 3) is true. Recall that Lτωn (F) denotes the lattice of all τ -closed n-multiply ω- composition subformations of F. In the case H = (1) from theorem 1 we obtain Corollary 1. Let F be a non-identity τ -closed n-multiply ω-composition formation (n ≥ 1). Then the following statements are equivalent: 1) Lτωn (F) is a complemented lattice; 2) F = ⊗i∈IFi, where {Fi \| i ∈ I} is the set of all atoms of the lattice Lτωn (F); 3) Lτωn (F) is a Boolean lattice. In the case when τ is trivial subgroup functor, n = 1, ω = P and H = (1) from theorem 1 we get 126 On cτωn -formations with Boolean sublattices Corollary 2 ([11], Theorem 2.3). Let F be a non-identity composition formation. Then the following statements are equivalent: 1) Lc(F) is a complemented lattice; 2) for any group G ∈ F, we have G = A×A1 × · · · ×At, where A is nilpotent and G, A1, . . . , At are simple non-abelian groups. Corollary 3 ([2], Theorem 6). Let F be a non-nilpotent n-multiply ω- composition formation (n ≥ 1). Then the following statements are equiva- lent: 1) F/ωnF ∩N is a complemented lattice; 2) F = (F ∩N) ∨ωn (∨ωnKi \| i ∈ I), where {Ki \| i ∈ I} is the set of all minimal n-multiply ω-composition non-nilpotent subformations of F; 3) F/ωnF ∩N is a Boolean lattice. Corollary 4 ([3], Theorem 1). Let F be a non-nilpotent ω-composition formation. Then the following statements are equivalent: 1) F/ωF ∩N is a complemented lattice; 2) F = (F ∩N) ∨ω (∨ωKi \| i ∈ I), where {Ki \| i ∈ I} is the set of all minimal ω-composition non-nilpotent subformations of F; 3) F/ωF ∩N is a Boolean lattice. References [1] A.N.Skiba, L.A.Shemetkov "Multiply L-composition formations of finite groups" [in Russian], Ukrainian Math. J., v. 52, No. 6, 2000, pp. 783-797. [2] P.A. Zhiznevsky "To the theory of multiply partially composition formations of finite groups" [in Russian], Preprint GGU im. F.Skoriny, 2008, No. 30, 35p. [3] P.A. Zhiznevsky, V.G.Safonov "On L-composition formations with complemented sublattices" [in Russian], Izvestiya vitebskogo gos. universiteta, 2008, No. 3(49), pp. 93-100. [4] L.A. Shemetkov, A.N. Skiba Formations of Algebraic Systems [in Russian], Nauka, Moscow, 1989. [5] A.N. Skiba, Algebra of Formations [in Russian], Bel. Navuka, Minsk, 1997. [6] G. Birkhoff "Lattice theory" New York: American mathematical society colloquium publications, Vol. XXV, 1948. [7] K. Doerk, T. Hawkes Finite Soluble Groups, Berlin; New York: Walter de Gruyter, 1992. [8] P.A. Zhiznevsky "On modularity and inductance of the lattice of all τ -closed n- multiply ω-composition finite groups formations" [in Russian], Izvestiya gomelskogo gos. universiteta, 2010, No. 1(58), pp. 185-191. [9] P.A. Zhiznevsky "On τ -closed n-multiply ω-composition formations with Boolean sublattice" [in Russian], Preprint GGU im. F.Skoriny, 2010, No. 3, 24p. [10] G. Grätzer "General lattice theory" New York: Academic Press, 1978. [11] I.V. Bliznets, A.N. Skiba "Critical and directly-reducible ω-composition formations" [in Russian], Preprint GGU im. F.Skoriny, 2002, No. 33, 23p. P. A. Zhiznevsky 127 Contact information P. A. Zhiznevsky Mathematics Department, Francisk Scorina Gomel State University, Sovetskaya Str., 104, 246019 Gomel, Belarus E-Mail: pzhiznevsky@yahoo.com Received by the editors: 07.02.2011 and in final form 26.02.2011.

On τ-closed n-multiply ω-composition formations with Boolean sublattices

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