Showing papers in "Algebra and Logic in 2000"
TL;DR: In this article, the authors give a description of δ-derivations of prime alternative and non-Lie Mal-tsev Φ-algebras, with some restrictions on an operator ring Φ.
Abstract: We give a description of δ-derivations of prime alternative and non-Lie Mal’tsev Φ-algebras, with some restrictions on an operator ring Φ. For algebras in these classes, every δ-derivation is proved trivial.
53 citations
TL;DR: In this paper, the concept of a minimal decomposition in a lattice is introduced and a new proof of the Crawley-Dilworth characterization theorem for the class of coalgebraic strongly coatomic lattices with unique irreducible decompositions is given.
Abstract: We define the concept of a minimal decomposition in a lattice, and prove that all the currently known lattices with unique irreducible decompositions are in fact lattices with minimal ones. Also, the characterization of a class of lattices with minimal decompositions is given. A new proof of the Crawley-Dilworth characterization theorem for the class of coalgebraic strongly coatomic lattices with unique irreducible decompositions obtains as a consequence.
46 citations
TL;DR: In this article, it was shown that a finite group whose element order set is the same as that of an alternating group An of degree n=r+1 or r+2 for prime r>5 or n=16 is isomorphic to An.
Abstract: It is proved that a finite group whose element order set is the same as that of an alternating group An of degree n=r+1 or r+2 for prime r>5 or n=16 is isomorphic to An.
39 citations
TL;DR: In this article, it was proved that a finite group isomorphic to a simple non-Abelian group up to isomorphism by a set of its element orders, and a list of all recognizable finite simple groups G for which 4t ∉ ω(G) with t>1 is known.
Abstract: It is proved that a finite group isomorphic to a simple non-Abelian group L3(2m) or U3(2m) is, up to isomorphism, recognizable by a set of its element orders. On the other hand, for every simple group S=S4(2m), there exist infinitely many pairwise non-isomorphic groups G with w(G)=w(S). As a consequence, we present a list of all recognizable finite simple groups G, for which 4t ∉ ω(G) with t>1.
37 citations
TL;DR: In this paper, two characterizations of projective linear groups PGL2(P) over a locally finite field P of characteristic 2 are defined in terms of permutation groups, and the second is defined by a structure of involution centralizers.
Abstract: The article contains two characterizations of projective linear groups PGL2(P) over a locally finite field P of characteristic 2: the first is defined in terms of permutation groups, and the second, in terms of a structure of involution centralizers. One of the two is used to prove the existence of infinite groups which are recognizable by the set of their element orders.
29 citations
TL;DR: In this article, it was shown that a group which contains elements of orders 1, 2, 3, 4, 5 and does not contain elements of any other order is locally finite and isomorphic either to an alternating group of degree 6 or to an extension of a nontrivial elementary Abelian 2-group by alternating groups of degree 5.
Abstract: We prove that a group which contains elements of orders 1, 2, 3, 4, 5 and does not contain elements of any other order is locally finite and isomorphic either to an alternating group of degree 6 or to an extension of a nontrivial elementary Abelian 2-group by an alternating group of degree 5.
22 citations
TL;DR: In this article, the authors studied Σ-definability in hereditarily finite superstructures over algebraic systems, and proved the Σ definability criterion, which was then used as a basis for establishing the reduction theorem for regular theories and obtaining a characterization of simple theories.
Abstract: Σ-definability in hereditarily finite superstructures over algebraic systems is studied. We prove the Σ-definability criterion, which is then used as a basis for establishing the reduction theorem for regular theories and for obtaining a characterization of simple theories. The idea of a nonstandard recursion theory is developed using subfields of the field of reals as an example. A partial algebraic description is given for a distributive upper semilattice of mΣ-degrees in hereditarily finite superstructures over models of simple theories.
15 citations
TL;DR: In this paper, the Cartesian degree of F2(A fixme n) is used to define a set of generators in the variety A fixme n of solvable groups and is universally equivalent to a free group Fr(A¯¯¯¯ n) in this variety.
Abstract: Every group that is finitely presented in the varietyA
n of solvable groups and is universally equivalent to a free group Fr(A
n) in this variety, is embedded in the Cartesian degree of F2(A
n) All subgroups on a set of two generators in that Cartesian degree which are universally equivalent to F2(A
n) are determined Free solvable and nilpotent groups are proved universally equivalent
14 citations
TL;DR: In this article, the authors define and study universal Horn classes dual to varieties in both the syntactic and semantic sense, called antivarieties, which appear naturally in graph theory and formal language theory.
Abstract: We define and study universal Horn classes dual to varieties in both the syntactic and the semantic sense. Such classes, which we call antivarieties, appear naturally, e.g., in graph theory and in formal language theory. The basic results are the characterization theorem for antivarieties, the theorem on cores in axiomatizable color-families, and the decidability theorem for universal theories of families of interpretations of formal languages.
13 citations
TL;DR: In this paper, it was shown that if G is a finite simple non-Abelian group, and N is some of its nilpotent subgroups, then |N|2<|G|.
Abstract: Orders and the structure of large nilpotent subgroups in all finite simple groups are determined. In particular, it is proved that if G is a finite simple non-Abelian group, and N is some of its nilpotent subgroups, then |N|2<|G|.
13 citations
TL;DR: In this article, the concept of reduced G-identities of rank n was introduced, which has a strong impact on the computations of coordinate groups for algebraic sets over groups.
Abstract: The fundamentals of algebraic geometry over a fixed group G were propounded in [1] where, in particular, the notion of a category of G-groups is introduced. Again. for groups in this category, the concepts of a G-identity and of a G-variety can be defined. We outline the groundwork for the theory of varieties in the category of G-groups. Most essential here is the idea of a group Vn,red(G) — of reduced G-identities of rank n. which has a strong impact on the computations of coordinate groups for algebraic sets over G. We prove that Vn,red(G)=1 for all natural n if G is a free-like or relatively free group for some variety of nilpotent groups whose rank is not less than the nilpotency class of G.
TL;DR: In this paper, it was shown that G is isomorphic to L2(Q) over a locally finite field Q of characteristic 2, and that the normalizer H =NG(S)=SλT is strongly embedded in G and is a Frobenius group with locally cyclic complement T.
Abstract: An involution i of a group G is said to be finite if |iig|<∞ for all g ∃ G. Suppose that G contains a finite involution and an infinite elementary Abelian 2-subgroup S and, moreover, the normalizer H=NG(S)=SλT is strongly embedded in G and is a Frobenius group with locally cyclic complement T. It is proved that G is isomorphic to L2(Q) over a locally finite field Q of characteristic 2. In particular, part (a) of Question 10.76 raised by Shunkkov in the Kourovka Notebook is answered in the affirmative.
TL;DR: In this paper, it was shown that the lattice of quasivarieties contained in a relatively subdirectly irreducible Sugihara semilattice with involution contains a sublattice isomorphic to a free lattice iff n ≥ 3.
Abstract: It is shown that the lattice of quasivarieties contained in the quasivariety generated by an n-element relatively subdirectly irreducible Sugihara semilattice with involution contains a sublattice isomorphic to the ideal lattice of a free lattice iff n≥3. Two consequences of the result are mentioned.
TL;DR: In this article, a purely algebraic algorithm is constructed for computing commutator length in a free group F2 (Thm. 1) with free generators a1, b1,..., ak, bk, k eN, every natural m satisfiescl(([a 1, b 1]... [ak, b k])m)=[(2 − m)/2] + mk.
Abstract: We study commutator length in free groups. (By a commutator lengthcl(g) of an element g in a derived subgroup G′ of a group G we mean the least natural number k such that g is a product of k commutators.) A purely algebraic algorithm is constructed for computing commutator length in a free group F2 (Thm. 1). Moreover, for every element z e F′2 and for any natural m, the following estimate derives:cl(zm) ≥ (ms(z) + 6)/12, where s(z) is a nonnegative number defined by an element z (Thm. 2). This estimate is used to compute commutator length of some particular elements. By analogy with the concept of width of a derived subgroup known in group theory, we define the concept of width of a derived subalgebra. The width of a derived subalgebra is computed for an algebra P of pairs, and also for its corresponding Lie algebra PL. The algebra of pairs arises naturally in proving Theorem 2 and enjoys a number of interesting properties. We state that in a free group F2k with free generators a1, b1, ..., ak, bk, k eN, every natural m satisfiescl(([a1, b1] ... [ak, bk])m)=[(2 − m)/2] + mk. For k=1, this entails a known result of Culler. The notion of a growth function as applied to a finitely generated group G is well known. Associated with a derived subgroup of F2 is some series depending on two variables which bears information not only on the number of elements of prescribed length but also on the number of elements of prescribed commutator length. A number of open questions are formulated.
TL;DR: In this article, the authors studied automorphism groups of Abelian groups G generated by quadratic automorphisms, that is, those of which each being an element of the endomorphism ring of G is a root of the Quadratic equation x2 + αx + β·1 with integral coefficients.
Abstract: We study automorphism groups of Abelian groups G generated by quadratic automorphisms, that is, those of which each being an element of the endomorphism ring of G is a root of the quadratic equation x2 + αx + β·1 with integral coefficients. Quadratic automorphisms are most notably exemplified by elements of orders 3 and 4 in groups of regular automorphisms: these are roots of the equations x2 + x + 1 and x2 + 1. respectively. Let A be generated by two quadratic automorphisms a and b of an Abelian group G. Then the following statements hold: (1) if the exponent m of G and the order n of ab are finite then A is a finite group of order at most m2n − 1; (2) if A is periodic then it is finite. Moreover, both of the finite conditions in (1) are essential. A consequence of these results is obtaining a description of periodic groups of regular automorphisms. generated by two automorphisms whose orders do not exceed 4.
TL;DR: In this article, the Baur-Monk-Garavaglia theorem on the elimination of quantifiers in the model theory of modules is generalized to a class of primitive connected theories, as distinct from modules.
Abstract: We prove the quantifier-elimination theorem for so-called primitive connected theories, exemplified by theories of modules. The theorem generalizes the well-known Baur-Monk-Garavaglia theorem on the elimination of quantifiers in the model theory of modules. The definition of a class of primitive connected theories, as distinct from modules. is not supposed to impose any conditions on a type of axioms that would specify those theories.
TL;DR: In this paper, it was shown that M =GP for some intermediate subring P (R ⊆ P ⊈ K) of a principal ideal ring R and GK of a Chevalley group over K.
Abstract: Let K be a field of fractions of a principal ideal ring R and GK be a Chevalley group (of normal type) over K. For each subring P ⊂ K, denote by GP a subgroup of all elements of GK with coefficients in P. Let M be intermediate between GR and GK, i.e., GR ⊆ M ⊆ GK. We prove that M=GP for some intermediate subring P (R ⊆ P ⊆ K).
TL;DR: In this article, the authors consider two properties which are close to being lower bounded in the class of finite join semidistributive lattices and construct an example of a finite join lattice with these properties, but it is not lower bounded.
Abstract: We consider two properties which are close to being lower bounded in the class of finite join semidistributive lattices. An example is constructed in which a finite join semidistributive lattice has both these two properties, but it is not lower bounded.
TL;DR: In this paper, it was shown that the class of rational subsets of a free nilpotent group G is a Boolean algebra iff G is Abelian-by-finite.
Abstract: We deal with a class of rational subsets of a group, that is, the least class of its subsets which contains all finite subsets and is closed under taking union. a product of two sets, and under generating of a submonoid by a set. It is proved that the class of rational subsets of a finitely generated nilpotent group G is a Boolean algebra iff G is Abelian-by-finite. We also study the question asking under which conditions the set of solutions for equations in groups will be rational. It is shown that the set of solutions for an arbitrary equation in one variable in a finitely generated nilpotent group of class 2 is rational. And we give an example of an equation in one variable in a free nilpotent group of nilpotency class 3 and rank 2 whose set of solutions is not rational.
TL;DR: In this paper, a complete regimentation for odd q is given, up to isomorphism, for q ≡ −1 (mod 4) and there is none for q ǫ > 0.
Abstract: We study groups\(\tilde H\). which are nonsplit extensions of elementary Abelian 2-groups A by H ∼ L2(q), with H acting on A irreducibly and A ≠ Z2. Related cohomology groups are computed. Groups\(\tilde H\) are given a complete regimentation for odd q. Moreover, there is only one, up to isomorphism, group\(\tilde H\) for q ≡ −1 (mod 4) and there is none for q ≡ 1 (mod 4). We also present an explicit construction of\(\tilde H\) treated as automorphism groups of some loops close to extraspecial groups, the so-called “code loops” brought in sight by Griess and Parker.
TL;DR: In this paper, it was shown that quasivarieties are closed under free products, and that each contains at most one maximal proper subquasivariariety.
Abstract: Let \(L(\mathcal{M})\) be a class of all groups G for which the normal closure (x)G of every element x belongs to a class \(\mathcal{M}\). \(L(\mathcal{M})\) is a Levi class generated by \(\mathcal{M}\). Let \(\mathcal{N}\) and \(\mathcal{N}\)0 be classes of finitely generated nilpotent groups and of torsion-free, finitely generated, nilpotent groups, respectively. We prove that \(q\mathcal{N}_0 \subset L(q\mathcal{N}_0 )\) and \(q\mathcal{N} \subset L(q\mathcal{N})\), and so \(L(q\mathcal{N}_0 )
e qL(\mathcal{N}_0 )\) and \(L(q\mathcal{N})
e qL(\mathcal{N})\). It is shown that quasivarieties \(L(q\mathcal{N})\) and \(L(q\mathcal{N}_0 )\) are closed under free products, and that each contains at most one maximal proper subquasivariety. It is also proved that \(L(\mathcal{M})\) is closed under free products if so is \(\mathcal{M}\).
TL;DR: In this paper, the authors re-prove the Makar-Limanov theorem on the existence of an algebraically closed skew field in the sense of there being a solution for any (generalized) polynomial equation.
Abstract: We re-prove the Makar-Limanov theorem on the existence of an algebraically closed skew field in the sense of there being a solution for any (generalized) polynomial equation. A new example of such a skew field is presented in which the Makar-Limanov construction is contained as a skew subfield. Our reasoning is underpinned by the main ideas of the original proof, but we employ a simpler argument for proving that the skew field constructed is algebraically closed.
TL;DR: This paper constructed a two-generated group with the co-recursively enumerable word problem that has no presentation by recursive permutations and exemplifies a group with a minimal possible number of generators.
Abstract: We construct a two-generated group with the co-recursively enumerable word problem that has no presentation by recursive permutations. This answers Higman’s question and exemplifies a group with the minimal possible number of generators. The previous article [1], in which that question was claimed settled, contains an incorrigible error.
TL;DR: In this article, it was proved that every countable ordinal lacks this property and a criterion for hereditarily finite admissible sets being existentially equivalent, which is interesting in its own right.
Abstract: We study into the question of which linearly ordered sets are intrinsically enumerable. In particular, it is proved that every countable ordinal lacks this property. To do this, we state a criterion for hereditarily finite admissible sets being existentially equivalent, which is interesting in its own right. Previously, Yu. L. Ershov presented the criterion for elements h
0
, h
1
in HF
$$\mathfrak{M}$$
) to realize a same type as applied to sufficiently saturated models
$$\mathfrak{M}$$
. Incidentally, that criterion fits with every model
$$\mathfrak{M}$$
on the condition that we limit ourselves to 1-types.
TL;DR: For any natural k ≥ 3 and l ≥ 2, the authors describes decidability boundaries for two types of groups: the variety of all k-nilpotent groups (KNP) and all l-solvable groups (LSP).
Abstract: For any natural k ≥ 3 and l ≥ 2, we describe decidability boundaries for two varieties: the variety of all k-nilpotent groups and the variety of all l-solvable groups.
TL;DR: In this paper, it was shown that if the lengths of all independent cycles of π are positive degrees of one number m ≥ 2 then n G π has a finite dimension equal to the number of prime divisors of m. The dimension of a variety is the least upper bound of lengths of independent bases for the collection of all strong Mal-tsev conditions satisfied in that variety.
Abstract: Let\(\mathbb{S}_n \) be a symmetric group on a set {1,2,...,n}. For an arbitrary permutation π of\(\mathbb{S}_n \), we consider a variety n G π ofn-groupoids (A, f) satisfying the identityf(x 1,x 2,...,x n )=f(x π(1),x π(2)...,x π(n)). It is proved that if lengths of all independent cycles of π are positive degrees of one numberm ≥2 then n G π has a finite dimension equal to the number of prime divisors ofm. The dimension of a variety, in this event, is the least upper bound of lengths of independent bases for the collection of all strong Mal’tsev conditions satisfied in that variety.
TL;DR: In this article, it was proved that there exists an incomplete Π10-degree which lacks cappings in enumeration Δ20-degrees, and that every low e-degree does have a capping in Δ20 e-decrees.
Abstract: We deal with a semilattice of Δ20-degrees, i.e., degrees that contain some set from a class Δ20 of the arithmetic hierarchy. It is proved that there exists an incomplete Π10-degree which lacks cappings in enumeration Δ20-degrees. On the other hand, it turns out that every low e-degree does have a capping in Δ20-e-degrees.
TL;DR: In this paper, a character table X of a finite group is broken up into four squares: A, B, C, and D, and relations via which ranks of the matrices in X are connected are established.
Abstract: A character table X of a finite group is broken up into four squares: A, B, C, and D. We establish relations via which ranks of the matrices inX are connected. In particular, ifX is an l × l-matrix, A is an s × t-matrix, and, moreover, the squares A and C are opposite, thenr(A)=r(C) + s + t − l; here.r(M) is the rank of a matrix M. Associated with such each block ofX is some integral nonnegative parameter m, and we have m=0 iff A, B, C, and D are active fragments ofX.
TL;DR: In this article, the authors consider the problem of proving Δ 1 1 -autostability of constructivizable Boolean algebras and show that the model is autostable if its Scott rank is strictly less than ω 1 CK.
Abstract: Let\(\mathfrak{M}\) be a Δ 1 1 -constructivizable model. If its Scott rank\(sr(\mathfrak{M})\) is strictly less than ω 1 CK , then it is proved autostable. But if\(sr(\mathfrak{M}) = \omega _1^{CK} \), then there exists an ordinal α α. We also consider some problems concerning Δ 1 1 -autostability of Δ 1 1 -constructivizable Boolean algebras.
TL;DR: In this paper, the elementary equivalence of lattices of subalgebras of free lattice varieties was shown to be equivalent to sets X and Y being second-order equivalent.
Abstract: A class of varieties V (including all finitely based lattice varieties) is determined for which the elementary equivalence of lattices of subalgebras of free V-algebras, Fv(X) and Fv(Y), is equivalent to sets X and Y being second-order equivalent.