Showing papers in "Algebra and Logic in 2007"
TL;DR: In this paper, the maximal tori of all finite simple classical groups, as well as of special and general projective linear and unitary groups, are treated, and their expression as a direct sum of cyclic groups is obtained in an explicit form.
Abstract: Maximal tori of all finite simple classical groups, as well as of special and general projective linear and unitary groups, are treated. For every such torus, its expression as a direct sum of cyclic groups is obtained in an explicit form.
66 citations
TL;DR: In this paper, the authors describe non-trivial δ-derivations of semisimple finite-dimensional Jordan algebras over an algebraically closed field of characteristic not 2.
Abstract: We describe non-trivial δ-derivations of semisimple finite-dimensional Jordan algebras over an algebraically closed field of characteristic not 2, and of simple finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic 0. For these classes of algebras and superalgebras, non-zero δ-derivations are shown to be missing for δ ≠ 0, 1/2, 1, and we give a complete account of 1/2-derivations.
43 citations
TL;DR: In this paper, the authors give a complete description of the structure of finite non-nilpotent groups all Schmidt subgroups of which are subnormal and give a description of their structure.
Abstract: We give a complete description of the structure of finite non-nilpotent groups all Schmidt subgroups of which are subnormal.
33 citations
TL;DR: In this paper, it was shown that the wreath product D ≀ A is an equationally Noetherian group and that free soluble groups of arbitrary derived lengths and ranks are equationally noetherian.
Abstract: Let B be a class of groups A which are soluble, equationally Noetherian, and have a central series A = A1 ⩾ A2 ⩾ … An ⩾ … such that ⋂An = 1 and all factors An/An+1 are torsion-free groups; D is a direct product of finitely many cyclic groups of infinite or prime orders. We prove that the wreath product D ≀ A is an equationally Noetherian group. As a consequence we show that free soluble groups of arbitrary derived lengths and ranks are equationally Noetherian.
24 citations
TL;DR: In this article, it was shown that the property of being locally constructivizable is inherited under Muchnik reducibility, which is weakest among the effective reducibilities considered over countable structures.
Abstract: We show that the property of being locally constructivizable is inherited under Muchnik reducibility, which is weakest among the effective reducibilities considered over countable structures. It is stated that local constructivizability of level higher than 1 is inherited under Σ-reducibility but is not inherited under Medvedev reducibility. An example of a structure
$$\mathfrak{M}$$
and a relation P ⊆ M is constructed for which
$${\underline {({\mathfrak{M}},P)}} \equiv {\underline {\mathfrak{M}}} $$
but
$$(\mathfrak{M},P)$$
≢∑
$$\mathfrak{M}$$
. Also, we point out a class of structures which are effectively defined by a family of their local theories.
23 citations
TL;DR: In this article, it is shown that the corresponding quotient algebra modulo the so-called h-equivalence is the simplest non-trivial semilattice with discrete closures.
Abstract: We introduce and study some natural operations on a structure of finite labeled forests, which is crucial in extending the difference hierarchy to the case of partitions. It is shown that the corresponding quotient algebra modulo the so-called h-equivalence is the simplest non-trivial semilattice with discrete closures. The algebra is also characterized as a free algebra in some quasivariety. Part of the results is generalized to countable labeled forests with finite chains.
17 citations
TL;DR: In this article, the authors argue for the existence of structures with the spectrum {x : x ≰ a} of degrees, where a is an arbitrary low degree, for any low degree a and b.
Abstract: We argue for the existence of structures with the spectrum {x : x ≰ a} of degrees, where a is an arbitrary low degree. Also it is stated that there exist structures with the spectrum of degrees, {x : x ≰ a} ⋃ {x : x ≰ b}, for any low degrees a and b.
17 citations
TL;DR: In this article, it was proved that the Carter subgroups of every finite almost simple group are conjugate, and based on their previous results, together with those obtained by F. Dalla Volta, A. Lucchini, and M. C. Tamburini, it was shown that the same class of subgroups also exist in the case of almost simple groups.
Abstract: In the paper we work to complete the classification of Carter subgroups in finite almost simple groups. In particular, it is proved that Carter subgroups of every finite almost simple group are conjugate. Based on our previous results, together with those obtained by F. Dalla Volta, A. Lucchini, and M. C. Tamburini, as a consequence we derive that Carter subgroups of every finite group are conjugate.
15 citations
TL;DR: In this article, the interpolation property in extensions of Johansson's minimal logic is investigated, and the construction of a matched product of models is proposed, which allows us to prove the interpolations property in a number of known extensions of the minimal logic.
Abstract: The interpolation property in extensions of Johansson’s minimal logic is investigated. The construction of a matched product of models is proposed, which allows us to prove the interpolation property in a number of known extensions of the minimal logic. It is shown that, unlike superintuitionistic, positive, and negative logics, a sum of J-logics with the interpolation property CIP may fail to possess CIP, nor even the restricted interpolation property.
13 citations
TL;DR: In this paper, the authors introduced the concept of Hochschild cohomologies for associative conformal algebras and proved that the second cohomology group of a conformal Weyl algebra with values in any bimodule is trivial.
Abstract: We introduce the concept of Hochschild cohomologies for associative conformal algebras. It is shown that the second cohomology group of a conformal Weyl algebra with values in any bimodule is trivial. As a consequence, we derive that the conformal Weyl algebra is segregated in any extension with nilpotent kernel.
13 citations
TL;DR: It is shown that, within the syntactic approach, any countable homogeneous model is generic, and criteria and a sufficient condition are given for the generic models created in syntactic constructions to be saturated.
Abstract: A syntactic approach is described to constructing generic models which generalizes the known semantic one. A sufficient condition of a generic model being homogeneous is specified. It is shown that, within the syntactic approach, any countable homogeneous model is generic. Criteria and a sufficient condition are given for the generic models created in syntactic constructions to be saturated.
TL;DR: In this article, the authors generalize Brauer, Suzuki, and Wall's result concerning the structure of finite groups with elementary Abelian centralizers of involutions to groups with almost perfect involutions.
Abstract: An involution i of a group G is said to be almost perfect in G if any two involutions of iG the order of a product of which is infinite are conjugated via a suitable involution in iG. We generalize a known result by Brauer, Suzuki, and Wall concerning the structure of finite groups with elementary Abelian centralizers of involutions to groups with almost perfect involutions.
TL;DR: In this article, the center of universal envelopes for Mal-tsev algebras is shown to be a ring of polynomials in a finite number of variables equal to the dimension of its Cartan subalgebra.
Abstract: Centers of universal envelopes for Mal’tsev algebras are explored. It is proved that the center of the universal envelope for a finite-dimensional semisimple Mal’tsev algebra over a field of characteristic 0 is a ring of polynomials in a finite number of variables equal to the dimension of its Cartan subalgebra, and that universal enveloping algebra is a free module over its center. Centers of universal enveloping algebras are computed for some Mal’tsev algebras of small dimensions.
TL;DR: A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group and a subgroup A is said to be semiautomormal in a group G if there exists an AB subgroup B such that AB = AB and AB1 is a proper subgroup of G, for every proper sub group B1 of B.
Abstract: A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB1 is a proper subgroup of G, for every proper subgroup B1 of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary.
TL;DR: In this paper, a paraconsistent extension of Sylvan's logic CCω is constructed, in which negation is defined via a total accessibility relation, and an axiomatization is given and the completeness theorem is proved.
Abstract: We deal with Sylvan’s logic CCω. It is proved that this logic is a conservative extension of positive intuitionistic logic. Moreover, a paraconsistent extension of Sylvan’s logic is constructed, which is also a conservative extension of positive intuitionistic logic and has the property of being decidable. The constructed logic, in which negation is defined via a total accessibility relation, is a natural intuitionistic analog of the modal system S5. For this logic, an axiomatization is given and the completeness theorem is proved.
TL;DR: In this article, the lattice of dominions is explored for down-semidistributivity, and it is shown that every quasivariety in this class contains an algebra whose lattice-of-domains is anti-isomorphic to a lattice.
Abstract: We fix a universal algebra A and its subalgebra H. The dominion of H in A (in a class M) is the set of all elements a ? A such that any pair of homomorphisms f, g: A ? M ? M satisfies the following: if f and g coincide on H then f(a) = g(a). In association with every quasivariety, therefore, is a dominion of H in A. Sufficient conditions are specified under which a set of dominions form a lattice. The lattice of dominions is explored for down-semidistributivity. We point out a class of algebras (including groups, rings) such that every quasivariety in this class contains an algebra whose lattice of dominions is anti-isomorphic to a lattice of subquasivarieties of that quasivariety.
TL;DR: In this paper, it was shown that for n ⩾ 2, the group G = \left\langle {G,x_1,x_2, \ldots,,x _n |w = 1} \right\rangle $$¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ always contains a non-Abelian free subgroup.
Abstract: Suppose that G is a non-trivial torsion-free group and w is a word over the alphabet G ⋃ {x
1
±1
,⋯,x
±1
}. It is proved that, for n ⩾ 2, the group
$$\tilde G = \left\langle {G,x_1 ,x_2 , \ldots ,,x_n |w = 1} \right\rangle $$
always contains a non-Abelian free subgroup. For n = 1, the question whether there exist non-Abelian free subgroups in
$$\tilde G$$
is amply settled for the unimodular case (i.e., where the exponent sum of x1 in w is one). Some generalizations of these results are discussed.
TL;DR: In this article, it was shown that the averaged Dehn function σ(n) is subasymptotic w.r.t. the function nl+1.
Abstract: It is proved that in any finite representation of any finitely generated nilpotent group of nilpotency class l ⩾ 1, the averaged Dehn function σ(n) is subasymptotic w.r.t. the function nl+1. As a consequence it is stated that in every finite representation of a free nilpotent group of nilpotency class l of finite rank r ⩾ 2, the Dehn function σ(n) is Gromov subasymptotic.
TL;DR: In this paper, the authors examined algebraic sets in a metabelian group G in two important cases: (1) G = Fn is a free metabelians group of rank n; (2) Wn,k is a wreath product of free Abelian groups of ranks n and k.
Abstract: The research launched in [1] is brought to a close by examining algebraic sets in a metabelian group G in two important cases: (1) G = Fn is a free metabelian group of rank n; (2) G = Wn,k is a wreath product of free Abelian groups of ranks n and k.
TL;DR: A complete description of (p, 1)-stable theories in terms of definable interpretability in a theory for the language of unary predicates is given in this article, where a definable interpretation of the theory is defined.
Abstract: A complete description of (p, 1)-stable theories is furnished in terms of definable interpretability in a theory for the language of unary predicates.
TL;DR: In this paper, it was shown that a periodic group G saturated with groups from the set {L3(2m)|m = 1, 2, …} is isomorphic to L3(Q), for a locally finite field Q of characteristic 2; in particular, it is locally finite.
Abstract: Let \(\mathfrak{M}\) be a set of finite groups. A group G is saturated with groups from \(\mathfrak{M}\) if every finite subgroup of G is contained in a subgroup isomorphic to some member of \(\mathfrak{M}\). It is proved that a periodic group G saturated with groups from the set {L3(2m)|m = 1, 2, …} is isomorphic to L3(Q), for a locally finite field Q of characteristic 2; in particular, it is locally finite.
TL;DR: In this article, the authors construct an example of a theory with a finite (greater than one) number of isomorphism types of countable models such that its prime and saturated models have computable presentations and there exists a model which lacks in such.
Abstract: We construct an example of a theory with a finite (greater than one) number of isomorphism types of countable models such that its prime and saturated models have computable presentations and there exists a model which lacks in such.
TL;DR: In this paper, it was shown that the semilattice of all c.e.r.t. m-degrees, from which the greatest element is removed, is isomorphic to the semi-attices of simple m-degrees, hypersimple m-diagrams, and Σ20-computable numberings of a finite family of sets.
Abstract: We deal with some upper semilattices of m-degrees and of numberings of finite families. It is proved that the semilattice of all c.e. m-degrees, from which the greatest element is removed, is isomorphic to the semilattice of simple m-degrees, the semilattice of hypersimple m-degrees, and the semilattice of Σ20-computable numberings of a finite family of Σ20-sets, which contains more than one element and does not contain elements that are comparable w.r.t. inclusion.
TL;DR: In this paper, the commutant of an FC-radical of a group G is shown to be finite and a normal nilpotent class 2 subgroup of finite index is found in G.
Abstract: An involution j of a group G is said to be almost perfect in G if any two involutions in jG whose product has infinite order are conjugated by a suitable involution in jG. Let G contain an almost perfect involution j and |CG(j)| < ∞. Then the following statements hold: (1) [j,G] is contained in an FC-radical of G, and |G: [j,G]| ⩽ |CG(j)|; (2) the commutant of an FC-radical of G is finite; (3) FC(G) contains a normal nilpotent class 2 subgroup of finite index in G.
TL;DR: In this paper, it was shown that any pair of irreducible characters of the group Sn sharing the same set of roots on one of the sets An and Sn \ An is divided into three parts.
Abstract: We show that treating of (non-trivial) pairs of irreducible characters of the group Sn sharing the same set of roots on one of the sets An and Sn \ An is divided into three parts. This, in particular, implies that any pair of such characters ?a and ?s (a and s are respective partitions of a number n) possesses the following property: lengths d(a) and d(s) of principal diagonals of Young diagrams for a and s differ by at most 1.
TL;DR: In this article, it was shown that the group of all computable automorphisms of the ordering of the rational numbers is its only model among the groups that are embeddable in all permutations.
Abstract: We prove that there is a first-order sentence ϕ such that the group of all computable automorphisms of the ordering of the rational numbers is its only model among the groups that are embeddable in the group of all computable permutations.
TL;DR: In this paper, it was shown that the existence of a supercompact cardinal is consistent with ZFC, and that the p-rank of Ext Ω(G, ℤ) is as large as possible for every prime p and for any torsion-free Abelian group G.
Abstract: We prove that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the p-rank of Ext
ℤ(G, ℤ) is as large as possible for every prime p and for any torsion-free Abelian group G. Moreover, given an uncountable strong limit cardinal µ of countable cofinality and a partition of Π (the set of primes) into two disjoint subsets Π0 and Π1, we show that in some model which is very close to ZFC, there is an almost free Abelian group G of size 2µ = µ+ such that the p-rank of Ext
ℤ(G, ℤ) equals 2µ = µ+ for every p ∈ Π0 and 0 otherwise, that is, for p ∈ Π1.
TL;DR: In this paper, the complexity of isomorphisms and relations on universes of structures, on the number and properties of numberings in various hierarchies of sets, and on the existence of connections between semantic and syntactic properties of the structures and relations for the class of nilpotent groups of class two.
Abstract: The paper deals in questions on the complexity of isomorphisms and relations on universes of structures, on the number and properties of numberings in various hierarchies of sets, and on the existence of connections between semantic and syntactic properties of the structures and relations for the class of nilpotent groups of class two.
TL;DR: This work deals with some issues on automatic recognition of interpolation properties in modal calculi extending the logics S5 and S4.
Abstract: We deal with some issues on automatic recognition of interpolation properties in modal calculi extending the logics S5 and S4.3.
TL;DR: In this paper, the authors considered infinite-dimensional locally soluble linear groups of infinite central dimension that are not soluble A3-groups and all of the proper subgroups of these groups have finite central dimension.
Abstract: We are concerned with infinite-dimensional locally soluble linear groups of infinite central dimension that are not soluble A3-groups and all of whose proper subgroups, which are not soluble A3-groups, have finite central dimension. The structure of groups in this class is described. The case of infinite-dimensional locally nilpotent linear groups satisfying the specified conditions is treated separately. A similar problem is solved for infinite-dimensional locally soluble linear groups of infinite fundamental dimension that are not soluble A3-groups and all of whose proper subgroups, which are not soluble A3-groups, have finite fundamental dimension.