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Том 58, № 4 (2019)
- Год: 2019
- Статей: 9
- URL: https://journal-vniispk.ru/0002-5232/issue/view/14556
Article
Sessions of the Seminar “Algebra i Logika”
366-367
Integral Cayley Graphs
Аннотация
Let G be a group and S ⊆ G a subset such that S = S−1, where S−1 = {s−1 | s ∈ S}. Then a Cayley graph Cay(G, S) is an undirected graph Γ with vertex set V (Γ) = G and edge set E(Γ) = {(g, gs) | g ∈ G, s ∈ S}. For a normal subset S of a finite group G such that s ∈ S ⇒ sk ∈ S for every k ∈ ℤ which is coprime to the order of s, we prove that all eigenvalues of the adjacency matrix of Cay(G, S) are integers. Using this fact, we give affirmative answers to Questions 19.50(a) and 19.50(b) in the Kourovka Notebook.
297-305
306-312
Universal Equivalence of Linear Groups Over Local Commutative Rings with 1/2
Аннотация
It is proved that the universal equivalence of general or special linear groups of orders greater than 2 over local commutative rings with 1/2 is equivalent to the coincidence of orders of groups and universal equivalence of respective rings.
313-321
Associators and Commutators in Alternative Algebras
Аннотация
It is proved that in a unital alternative algebra A of characteristic ≠ 2, the associator (a, b, c) and the Kleinfeld function f(a, b, c, d) never assume the value 1 for any elements a, b, c, d ∈ A. Moreover, if A is nonassociative, then no commutator [a, b] can be equal to 1. As a consequence, there do not exist algebraically closed alternative algebras. The restriction on the characteristic is essential, as exemplified by the Cayley–Dickson algebra over a field of characteristic 2.
322-326
A Levi Class Generated by a Quasivariety of Nilpotent Groups
Аннотация
Let L(M) be a class of all groups G in which the normal closure of any element belongs to M; qM is a quasivariety generated by a class M. We consider a quasivariety qH2 generated by a relatively free group in a class of nilpotent groups of class at most 2 with commutator subgroup of exponent 2. It is proved that the Levi class L(qH2) generated by the quasivariety qH2 is contained in the variety of nilpotent groups of class at most 3.
327-336
Asymptotic Rank Theorems
Аннотация
Let A be a numerical k ×∞-matrix such that minors AI of order k tend to zero if numbers of all columns forming these minors tend to infinity. It is shown that there exits a nontrivial linear combination of rows in A which is a sequence tending to zero.
337-344
Prime and Homogeneous Rings and Algebras
Аннотация
Let ℳ be a structure of a signature Σ. For any ordered tuple \( \overline{a}=\left({a}_1,\dots, {a}_{\mathrm{n}}\right) \) of elements of ℳ, \( {\mathrm{tp}}^{\mathcal{M}}\left(\overline{a}\right) \) denotes the set of formulas θ(x1, …, xn) of a first-order language over Σ with free variables x1, . . . , xn such that \( \mathcal{M}\left|=\theta \left({a}_1,\dots, {a}_n\right)\right. \). A structure ℳ is said to be strongly ω-homogeneous if, for any finite ordered tuples \( \overline{a} \) and \( \overline{b} \) of elements of ℳ, the coincidence of \( {\mathrm{tp}}^{\mathcal{M}}\left(\overline{a}\right) \) and \( {\mathrm{tp}}^{\mathrm{M}}\left(\overline{b}\right) \) implies that these tuples are mapped into each other (componentwise) by some automorphism of the structure ℳ. A structure ℳ is said to be prime in its theory if it is elementarily embedded in every structure of the theory Th (ℳ). It is proved that the integral group rings of finitely generated relatively free orderable groups are prime in their theories, and that this property is shared by the following finitely generated countable structures: free nilpotent associative rings and algebras, free nilpotent rings and Lie algebras. It is also shown that finitely generated non-Abelian free nilpotent associative algebras and finitely generated non-Abelian free nilpotent Lie algebras over uncountable fields are strongly ω-homogeneous.
345-355
Khutoretskii’s Theorem for Generalized Computable Families
Аннотация
We give sufficient conditions for generalized computable numberings to satisfy the statement of Khutoretskii’s theorem. This implies limitedness of universal \( {\varSigma}_{\alpha}^0- \) computable numberings for 2 \( \le \alpha <{\omega}_1^{CK}. \)
356-365