Algebra and Logic最新文献

According to G. Birkhoff, there is a categorical duality between the category of bi-algebraic distributive (0, 1)-lattices with complete (0, 1)-lattice homomorphisms as morphisms and the category of partially ordered sets with partial order-preserving maps as morphisms. We extend this classical result to the bi-algebraic lattices belonging to the variety of (0, 1)-lattices generated by the pentagon, the 5-element nonmodular lattice. Applying the extended duality, we prove that the lattice of quasivarieties contained in the variety of (0, 1)-lattices generated by the pentagon has uncountably many elements and is not distributive. This yields the following: the lattice of quasivarieties contained in a nontrivial variety of (0, 1)-lattices either is a 2-element chain or has uncountably many elements and is not distributive.

We generalize the constructions of Q- and G-families of quandles introduced in the paper of A. Ishii et al. in [Ill. J. Math., 57, No. 3, 817-838 (2013)], and establish how they are related to other constructions of quandles. A composition of structures of quandles defined on the same set is specified, and conditions are found under which this composition yields a quandle. It is proved that under such a multiplication we obtain a group that will be Abelian. Also a direct product of quandles is examined.

We study periodic groups saturated with finite simple symplectic groups.

For a finite group G, we denote by N (G) the set of its conjugacy class sizes. Recently, the following question was posed: given any n ∈ ℕ and an arbitrary non-Abelian finite simple group S, is it true that G ≃ Sⁿ if G is a group with trivial center and N (G) = N (Sⁿ)? The answer to this question is known for all simple groups S with n = 1, and also for S ∈ {A₅, A₆}, where A_k denotes the alternating group of degree k, with n = 2. It is proved that the group A₅ × A₅ × A₅ is uniquely defined by the set N (A₅ × A₅ × A₅) in the class of finite groups with trivial center.

We study degrees and degree spectra of groups ({mathfrak{G}}_{mathrm{I}}) defined on a set of permutations of the natural numbers ω whose degrees belong to a Turing ideal I. A necessary condition and a sufficient condition are stated which specify whether an arbitrary Turing degree belongs to the degree spectrum of a group ({mathfrak{G}}_{mathrm{I}}). Nonprincipal ideals I for which the group ({mathfrak{G}}_{mathrm{I}}) has or does not have a degree are exemplified.

A variety of associative algebras is nonmatrix if it does not contain the algebra of 2 × 2 matrices over a given field. Nonmatrix varieties were introduced and studied by V. N. Latyshev in [Algebra and Logic, 16, No. 2, 98-122 (1977); Algebra and Logic, 16, No. 2, 122-133 (1977); Mat. Zam., 27, No. 1, 147-156 (1980)] in connection with the Specht problem. A series of equivalent characterizations of nonmatrix varieties was obtained in [Isr. J. Math., 181, No. 1, 337-348 (2011)]. In the present paper, the notion of nonmatrix variety is extended to nonassociative algebras, and their characterization from the last-mentioned paper is generalized to alternative, Jordan, and some other varieties of algebras.

We introduce a bipolar classification with index j for endomorphisms of an arbitrary n-groupoid with n > 1, where j = 1, 2, . . . , n. The classifications of endomorphisms constructed generalize the bipolar classification of endomorphisms of an arbitrary groupoid (i.e., a 2-groupoid) introduced previously. Using a left bipolar classification of endomorphisms of an n-groupoid (a particular case of the obtained classifications), we succeed in constructing an integral classification of endomorphisms of an arbitrary algebra (i.e., a structure without relations) with finitary operations.

We will look into the following conjecture, which, if valid, would allow us to formulate an unimprovable analog of the Baer–Suzuki theorem for the π-radical of a finite group (here π is an arbitrary set of primes). For an odd prime number r, put m = r, if r = 3, and m = r - 1 if r ≥ 5. Let L be a simple non-Abelian group whose order has a prime divisor s such that s = r if r divides |L| and s > r otherwise. Suppose also that x is an automorphism of prime order of L. Then some m conjugates of x in the group (langle L,xrangle ) generate a subgroup of order divisible by s. The conjecture is confirmed for the case where L is a group of Lie type and x is an automorphism induced by a unipotent element.

A finite Frobenius group in which the order of complements is divisible by a prime number p is called a Φ_p-group. We prove the theorem stating the following. Let G be a periodic group with a finite element a of prime order p > 2 saturated with Φ_p-groups. Then G = F λ H is a Frobenius group with kernel F and complement H. If G contains an involution i commuting with the element a, then H = C_G(i) and F is Abelian, and H = N_G((langle arangle )) otherwise.