It is proved that the lattice of all τ-closed totally saturated formations of finite groups is al... more It is proved that the lattice of all τ-closed totally saturated formations of finite groups is algebraic. In the paper we consider finite groups only. Information on all relevant terms is contained in [1-4]. The methods and constructions of general lattice theory used in exploring the inner structure of formations allow of creating simpler schemes of proving both the known facts and new results in formation theory (see [2-5]). General properties of the lattice of totally saturated formations as well as the structure of such formations with prescribed restrictions on lattices of totally saturated subformations are dealt with in [6-13]. At the same time the lattice of totally saturated formations is now one of the least known lattices of group formations. A number of open questions posed in [2-5, 14] testify to this judgement. In [3], it was proved that for any non-negative integer n, both the lattice of all τ-closed n-multiply saturated formations and the lattice of all soluble totally saturated ones are algebraic. Moreover, the question was posed whether the lattice l τ ∞ of all τ-closed totally saturated formations is algebraic (see [3, Question 4.4.6]). In the present paper, this question is answered in the affirmative. We recall some of the notation and definitions. A non-empty system θ of formations is called a complete lattice of formations if the intersection of any family of formations in θ is again in θ and the set θ contains a formation F such that H ⊆ F, for any formation H ∈ θ. The formations in θ are referred to as θ-formations. Let A and B be groups, ϕ : A → B be an epimorphism, and Ω and Σ be some systems of subgroups in A and B, respectively. We denote by Ω ϕ the set {H ϕ | H ∈ Ω}, and by Σ ϕ −1 the set {H ϕ −1 | H ∈ Σ} of all preimages in A of all groups of Σ. Let X be any non-empty class of groups, and let G ∈ X be associated with some system τ (G) of its subgroups. Following [3] we say that τ is a subgroup X-functor (or else τ is a subgroup functor on X) if (τ (A)) ϕ ⊆ τ (B) and (τ (B)) ϕ −1 ⊆ τ (A) for every epimorphism ϕ : A → B, where A, B ∈ X; moreover, G ∈ τ (G) for any group G ∈ X. The class F of groups is said to be τ-closed if τ (G) ⊆ F, for any group G ∈ F. Every formation of finite groups is said to be 0-multiply saturated. For n 1, the formation F is n-multiply saturated if it has a local screen such that all non-empty values of the screen are (n − 1)-multiply saturated formations. A formation that is n-multiply saturated for any non-negative integer n is referred to as totally saturated. If, in addition, F is τ-closed then we call F a τ-closed n-multiply saturated, and accordingly, a τ-closed totally saturated, formation.
References 1. V. D. Belousov, Foundations of the theory of quasigroups and loops (in Russian). Mo... more References 1. V. D. Belousov, Foundations of the theory of quasigroups and loops (in Russian). Moscow, „Nauka", 1967. 2. C. Lindner, D. Steedly, On the number of conjugates of a quasigroup. Algebra Univ. 5 (1975), p. 191-196. 3. T. Popovich, On conjugate sets of quasigroups. Bul. Acad. de Stiinte a Republicii Moldova. Matematica, N1 (59), 2012, p. 21-32. 4. G. Belyavskaya, T. Popovich, Conjugate sets of loops and quasigroups. DC-quasigroups. Bul. Acad. de Stiinte a Republicii Moldova. Matematica, N1 (68), 2012, p. 21-31.
При исследовании и классификации формаций конечных групп возникает необходимость изучения их внут... more При исследовании и классификации формаций конечных групп возникает необходимость изучения их внутреннего строения, опираясь на свойства некоторых хорошо изученных классов групп. Важным инструментом таких исследований являются критические формации, т.е. минимальные по включению формации, не содержащиеся в заданном классе групп, все собственные подформации которых в нем содержатся. В частности, свойства решетки подформаций исследуемой формации тесно связаны с наличием или отсутствием критических подформаций того или иного вида, а также от их взаимного расположения в данной формации. Актуальной является задача развития метода критических формаций для изучения частично тотально насыщенных формаций конечных групп. Цель исследования-описание минимальных тотально ω-насыщенных ненильпотентных формаций конечных групп. Материал и методы. Материалом для исследования является решетка тотально ω-насыщенных подформаций то-тально ω-насыщенной формации конечных групп. Используются методы теории кон...
In the universe of finite groups the description of τ-closed totally saturated formations with Bo... more In the universe of finite groups the description of τ-closed totally saturated formations with Boolean sublattices of τ-closed totally saturated subformations is obtained. Thus, we give a solution of Question 4.3.16 proposed by A. N. Skiba in his monograph "Algebra of Formations" (1997).
In the universe of finite groups the description of �-closed totally saturated formations with Bo... more In the universe of finite groups the description of �-closed totally saturated formations with Boolean sublattices of �-closed totally saturated subformations is obtained. Thus, we give a solution of Question 4.3.16 proposed by A. N. Skiba in his monograph "Algebra of Formations" (1997).
It is proved that the lattice of �-closed totally saturated formations of finite groups is distri... more It is proved that the lattice of �-closed totally saturated formations of finite groups is distributive. This is a solu- tion of Question 4. 2. 15 proposed by A. N. Skiba in his monograph "Algebra of Formations" (1997).
We prove that the finite length condition for the lattice of τ-closed totally saturated subformat... more We prove that the finite length condition for the lattice of τ-closed totally saturated subformations of a τ-closed totally saturated formation is equivalent to the finite length condition for this lattice, and it is also equivalent for the formation to be soluble and one-generated.
It is proved that the lattice of totally saturated formations of finite groups is distributive. T... more It is proved that the lattice of totally saturated formations of finite groups is distributive. Thus, we give an affirmative answer to the problem proposed by Shemetkov, Skiba and Guo.
It is proved that the lattice of all τ-closed totally saturated formations of finite groups is G-... more It is proved that the lattice of all τ-closed totally saturated formations of finite groups is G-separable. All groups considered in the paper are assumed to be finite. We adhere to the terminology adopted in [1-4]. Research into the inner structure of formations of various types and regimentation of such formations are an intensively evolving direction in formation theory. An essential part in so doing is played by methods and constructions employed in general lattice theory, which allows properties of naturally arising lattices of formations to be used and elegant and most compact proof schemes to be created (see [2-5]). A number of properties of the lattice of all totally saturated formations were established in [3, 6-14]. In this account, we will look at properties of the lattice l τ ∞ of all τ-closed totally saturated formations which are related to the concept of being X-separable for a lattice of formations. Let X be some nonempty class of groups. A complete lattice θ of formations is said to be
It is proved that the lattice of all τ-closed totally saturated formations of finite groups is al... more It is proved that the lattice of all τ-closed totally saturated formations of finite groups is algebraic. In the paper we consider finite groups only. Information on all relevant terms is contained in [1-4]. The methods and constructions of general lattice theory used in exploring the inner structure of formations allow of creating simpler schemes of proving both the known facts and new results in formation theory (see [2-5]). General properties of the lattice of totally saturated formations as well as the structure of such formations with prescribed restrictions on lattices of totally saturated subformations are dealt with in [6-13]. At the same time the lattice of totally saturated formations is now one of the least known lattices of group formations. A number of open questions posed in [2-5, 14] testify to this judgement. In [3], it was proved that for any non-negative integer n, both the lattice of all τ-closed n-multiply saturated formations and the lattice of all soluble totally saturated ones are algebraic. Moreover, the question was posed whether the lattice l τ ∞ of all τ-closed totally saturated formations is algebraic (see [3, Question 4.4.6]). In the present paper, this question is answered in the affirmative. We recall some of the notation and definitions. A non-empty system θ of formations is called a complete lattice of formations if the intersection of any family of formations in θ is again in θ and the set θ contains a formation F such that H ⊆ F, for any formation H ∈ θ. The formations in θ are referred to as θ-formations. Let A and B be groups, ϕ : A → B be an epimorphism, and Ω and Σ be some systems of subgroups in A and B, respectively. We denote by Ω ϕ the set {H ϕ | H ∈ Ω}, and by Σ ϕ −1 the set {H ϕ −1 | H ∈ Σ} of all preimages in A of all groups of Σ. Let X be any non-empty class of groups, and let G ∈ X be associated with some system τ (G) of its subgroups. Following [3] we say that τ is a subgroup X-functor (or else τ is a subgroup functor on X) if (τ (A)) ϕ ⊆ τ (B) and (τ (B)) ϕ −1 ⊆ τ (A) for every epimorphism ϕ : A → B, where A, B ∈ X; moreover, G ∈ τ (G) for any group G ∈ X. The class F of groups is said to be τ-closed if τ (G) ⊆ F, for any group G ∈ F. Every formation of finite groups is said to be 0-multiply saturated. For n 1, the formation F is n-multiply saturated if it has a local screen such that all non-empty values of the screen are (n − 1)-multiply saturated formations. A formation that is n-multiply saturated for any non-negative integer n is referred to as totally saturated. If, in addition, F is τ-closed then we call F a τ-closed n-multiply saturated, and accordingly, a τ-closed totally saturated, formation.
References 1. V. D. Belousov, Foundations of the theory of quasigroups and loops (in Russian). Mo... more References 1. V. D. Belousov, Foundations of the theory of quasigroups and loops (in Russian). Moscow, „Nauka", 1967. 2. C. Lindner, D. Steedly, On the number of conjugates of a quasigroup. Algebra Univ. 5 (1975), p. 191-196. 3. T. Popovich, On conjugate sets of quasigroups. Bul. Acad. de Stiinte a Republicii Moldova. Matematica, N1 (59), 2012, p. 21-32. 4. G. Belyavskaya, T. Popovich, Conjugate sets of loops and quasigroups. DC-quasigroups. Bul. Acad. de Stiinte a Republicii Moldova. Matematica, N1 (68), 2012, p. 21-31.
При исследовании и классификации формаций конечных групп возникает необходимость изучения их внут... more При исследовании и классификации формаций конечных групп возникает необходимость изучения их внутреннего строения, опираясь на свойства некоторых хорошо изученных классов групп. Важным инструментом таких исследований являются критические формации, т.е. минимальные по включению формации, не содержащиеся в заданном классе групп, все собственные подформации которых в нем содержатся. В частности, свойства решетки подформаций исследуемой формации тесно связаны с наличием или отсутствием критических подформаций того или иного вида, а также от их взаимного расположения в данной формации. Актуальной является задача развития метода критических формаций для изучения частично тотально насыщенных формаций конечных групп. Цель исследования-описание минимальных тотально ω-насыщенных ненильпотентных формаций конечных групп. Материал и методы. Материалом для исследования является решетка тотально ω-насыщенных подформаций то-тально ω-насыщенной формации конечных групп. Используются методы теории кон...
In the universe of finite groups the description of τ-closed totally saturated formations with Bo... more In the universe of finite groups the description of τ-closed totally saturated formations with Boolean sublattices of τ-closed totally saturated subformations is obtained. Thus, we give a solution of Question 4.3.16 proposed by A. N. Skiba in his monograph "Algebra of Formations" (1997).
In the universe of finite groups the description of �-closed totally saturated formations with Bo... more In the universe of finite groups the description of �-closed totally saturated formations with Boolean sublattices of �-closed totally saturated subformations is obtained. Thus, we give a solution of Question 4.3.16 proposed by A. N. Skiba in his monograph "Algebra of Formations" (1997).
It is proved that the lattice of �-closed totally saturated formations of finite groups is distri... more It is proved that the lattice of �-closed totally saturated formations of finite groups is distributive. This is a solu- tion of Question 4. 2. 15 proposed by A. N. Skiba in his monograph "Algebra of Formations" (1997).
We prove that the finite length condition for the lattice of τ-closed totally saturated subformat... more We prove that the finite length condition for the lattice of τ-closed totally saturated subformations of a τ-closed totally saturated formation is equivalent to the finite length condition for this lattice, and it is also equivalent for the formation to be soluble and one-generated.
It is proved that the lattice of totally saturated formations of finite groups is distributive. T... more It is proved that the lattice of totally saturated formations of finite groups is distributive. Thus, we give an affirmative answer to the problem proposed by Shemetkov, Skiba and Guo.
It is proved that the lattice of all τ-closed totally saturated formations of finite groups is G-... more It is proved that the lattice of all τ-closed totally saturated formations of finite groups is G-separable. All groups considered in the paper are assumed to be finite. We adhere to the terminology adopted in [1-4]. Research into the inner structure of formations of various types and regimentation of such formations are an intensively evolving direction in formation theory. An essential part in so doing is played by methods and constructions employed in general lattice theory, which allows properties of naturally arising lattices of formations to be used and elegant and most compact proof schemes to be created (see [2-5]). A number of properties of the lattice of all totally saturated formations were established in [3, 6-14]. In this account, we will look at properties of the lattice l τ ∞ of all τ-closed totally saturated formations which are related to the concept of being X-separable for a lattice of formations. Let X be some nonempty class of groups. A complete lattice θ of formations is said to be
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