Algebra and Logic最新文献

We prove that the wreath product C = A ≀ B of a semigroup A with zero and an infinite cyclic semigroup B is q_ω-compact (logically Noetherian). Our result partially solves I. Plotkin‘s problem for wreath products.

It is proved that among finite simple non-Abelian groups only the groups U₃(3) and A₈ are not generated by three conjugate involutions. This result is obtained modulo a known conjecture on the description of finite simple groups generated by two elements of orders 2 and 3.

A topological duality result is established for the category of distributive c-posets defined in this paper, as well as for some of its important full subcategories. All duality results presented extend the well-known topological duality result obtained by M. H. Stone for the category of distributive (0, 1)-lattices.

Using the functional approach of R. Baer and B. I. Plotkin, we introduce and study the notion of ℱ-functorial whose values are characteristic subgroups of a finite group that possess certain properties of the Fitting subgroup. The lattice and semigroups of ℱ-functorials are described, the interrelation between ℱ-functorials and classes of groups is established, a characterization of their values is given in terms of group’s elements inducing inner automorphisms on specified chief factors.

We look at the interconnection between Lie nilpotent Jordan algebras and Lie nilpotent associative algebras. It is proved that a special Jordan algebra is Lie nilpotent if and only if its associative enveloping algebra is Lie nilpotent. Also it turns out that a Jordan algebra is Lie nilpotent of index 2n + 1 if and only if its algebra of multiplications is Lie nilpotent of index 2n. Finally, we prove a product theorem for Jordan algebras.

Let R and R^φ be associative rings with isomorphic subring lattices, and φ be a lattice isomorphism (or else a projection) of the ring R onto the ring R^φ. We call R^φ the projective image of a ring R and call R itself the projective preimage of a ring R^φ. The main result of the first part of the paper is Theorem 5, which proves that the projective image R^φ of a one-generated finite p-ring R is also one-generated if R^φ at the same time is itself a p-ring. In the second part, we continue studying projections of matrix rings. The main result of this part is Theorems 6 and 7, which prove that if R = M_n(K) is the ring of all square matrices of order n over a finite ring K with identity, and φ is a projection of the ring R onto the ring R^φ, then R^φ = M_n(K′), where K′ is a ring with identity, lattice-isomorphic to the ring K.

G. Malle, J. Saxl, and T. Weigel in [Geom. Ded., 49, No. 1, 85-116 (1994)] formulated the following problem: For every finite simple non-Abelian group G, find the minimum number n_c(G) of generators of conjugate involutions whose product equals 1. (See also Question 14.69c in [Unsolved Problems in Group Theory. The Kourovka Notebook, No. 20, E. I. Khukhro and V. D. Mazurov (Eds.), Sobolev Institute of Mathematics SO RAN, Novosibirsk (2022); https://alglog.org/20tkt.pdf].) J. M. Ward [PhD Thesis, Queen Mary College, Univ. London (2009)] solved this problem for sporadic, alternating, and projective special linear groups PSL_n(q) over a field of odd order q, except in the case q = 9 for n ≥ 4 and also in the case q ≡ 3 (mod 4) for n = 6. Here we lift the restriction q ≠ 9 for dimensions n ≥ 9 and n = 6.

Let σ be a partition of the set of all prime numbers into a union of pairwise disjoint subsets. Using the idea of multiple localization due to A. N. Skiba, we introduce the notion of a Baer n-multiply σ-local formation of finite groups. It is proved that with respect to inclusion ⊆, the collection of all such formations form a complete algebraic modular lattice. Thereby we generalize the result obtained by A. N. Skiba and L. A. Shemetkov in [Ukr. Math. J., 52, No. 6, 783-797 (2000)].

Lattices are considered in which, instead of distributive identities, a ‘gap’ of length at most 1 is allowed between the right and left parts of each distributivity relation. Such lattices are said to be close to distributive ones. Although this property is weaker than distributivity, nevertheless a 3-generated lattice with this property is also finite.