Torsion-free Abelian Research Articles

Further results on generalized cellular automata

Given a finite set A and a group homomorphism ϕ : H → G , a ϕ -cellular automaton is a function T : A G → A H that is continuous with respect to the prodiscrete topologies and ϕ -equivariant in the sense that h · T ( x ) = T ( ϕ ( h ) · x ) , for all x ∈ A G , h ∈ H , where · denotes the shift actions of G and H on A G and AH , respectively. When G = H and ϕ = id , the definition of id -cellular automata coincides with the classical definition of cellular automata. The purpose of this paper is to expand the theory of ϕ -cellular automata by focusing on the differences and similarities with their classical counterparts. After discussing some basic results, we introduce the following definition: a ϕ -cellular automaton T : A G → A H has the unique homomorphism property (UHP) if T is not ψ-equivariant for any group homomorphism ψ : H → G , ψ ≠ ϕ . We show that if the difference set Δ ( ϕ , ψ ) is infinite, then T is not ψ-equivariant; it follows that when G is torsion-free abelian, every non-constant T has the UHP. Furthermore, inspired by the theory of classical cellular automata, we study ϕ -cellular automata over quotient groups, as well as their restriction and induction to subgroups and supergroups, respectively.

Algorithms in Direct Decompositions of Torsion-Free Abelian Groups

Algorithms in Direct Decompositions of Torsion-Free Abelian Groups

Matrix Representations of Endomorphism Rings for Torsion-Free Abelian Groups

Matrix Representations of Endomorphism Rings for Torsion-Free Abelian Groups

Frattini-resistant direct products of pro-p groups

A pro-p group G is called strongly Frattini-resistant if the function H↦Φ(H), from the poset of all closed subgroups of G into itself, is a poset embedding. Frattini-resistant pro-p groups appear naturally in Galois theory. Indeed, every maximal pro-p Galois group over a field that contains a primitive pth root of unity (and also contains −1 if p=2) is strongly Frattini-resistant. Let G1 and G2 be non-trivial pro-p groups. We prove that G1×G2 is strongly Frattini-resistant if and only if one of the direct factors G1 or G2 is torsion-free abelian and the other one has the property that all of its closed subgroups have torsion-free abelianization. As a corollary we obtain a group theoretic proof of a result of Koenigsmann on maximal pro-p Galois groups that admit a non-trivial decomposition as a direct product. In addition, we give an example of a group that is not strongly Frattini-resistant, but has the property that its Frattini-function defines an order self-embedding of the poset of all topologically finitely generated subgroups.

Influence of the Baer–Kaplansky Theorem on the Development of the Theory of Groups, Rings, and Modules

The review presents an analysis of results of the Abelian group theory, as well as rings and modules, which concern the definability of algebraic structures by their endomorphism rings and related structures. In the systematization of the results, the greatest attention is paid to torsion-free Abelian groups, which are of particular interest due to the presence of non-isomorphic direct decompositions in this class. This significantly expands the understanding of general, including modern, trends of the development of algebra in the context related to the Baer–Kaplansky theorem. The reflection of the properties of algebraic objects of a certain class in their endomorphism rings is a natural structural connection, the study of which is a separate investigation direction. A striking introduction to this topic was the Baer–Kaplansky theorem for torsion Abelian groups, which dates back to the middle of the last century and states that any isomorphism of endomorphism rings of two groups from this class is inevitably induced by some isomorphism of the groups themselves. Of course, it follows that if two torsion Abelian groups have isomorphic endomorphism rings, then they are isomorphic. This profound result inspired mathematicians to obtain results in the same form concerning other classes of objects. But even in the theory of Abelian groups itself, other classes were discovered for which the analogue of the Baer–Kaplansky theorem is valid. Despite the fundamental difference between the definitions of completely decomposable Abelian groups, which are direct sums of rank-one torsion-free groups, and torsion Abelian groups considered, which are direct sums of finite order cyclic groups, there is one very important common characteristic of these classes: their decompositions into indecomposable summands are determined uniquely up to isomorphism. This property is not possessed by torsion-free Abelian groups in general, whose definability by their endomorphism rings is in the focus of our attention.

Solving logistic tasks by parallelizing algorithms of the theory of direct decompositions of torsion-free abelian groups

The paper considers the principles of parallelization at marshalling yards and determines their importance. There are presented the methods for direct decompositions of torsion-free Abelian groups of finite rank. There are applied the concepts of “conditionally linear graphs” depicting railway trains and their characteristics, including “quasi-length”, which characterizes the length of the corresponding “quasi-graph”. There has been formalized a graphical approach to the redistribution of railcars of incoming trains at a marshalling yard for the formation of outgoing trains. The corresponding algorithms have been developed.

Applications of analytic number theory to the study of type sets of torsionfree Abelian groups. II.

Applications of analytic number theory to the study of type sets of torsionfree Abelian groups. II.

Applications of analytic number theory to the study of type sets of torsionfree Abelian groups. I.

Applications of analytic number theory to the study of type sets of torsionfree Abelian groups. I.

On transitivity-like properties for torsion-free Abelian groups

Abstract We study some close relationships between the classes of transitive, fully transitive and Krylov transitive torsion-free Abelian groups. In addition, as an application of the achieved assertions, we resolve some old-standing problems, posed by Krylov, Mikhalev and Tuganbaev in their monograph [P. A. Krylov, A. V. Mikhalev and A. A. Tuganbaev, Endomorphism Rings of Abelian Groups, Kluwer Academic, Dordrecht, 2003]. Specifically, we answer Problem 44 from there in the affirmative by constructing a Krylov transitive torsion-free Abelian group which is neither fully transitive nor transitive. This extends to the torsion-free case certain similar results in the p-torsion case. We, alternatively, also expand to the torsion-free version some of the results concerning transitivity, full transitivity and Krylov transitivity in the p-primary case.

A Feasible Algorithm for Solubility of Transitive Graphs

Transitive, weighted graphs, with the weights coming from an Abelian group and satisfying a transitivity property, were studied in [1] and were referred to as the transitive systems. Of central importance is the question of completing a given system to a transitive system where every ordered pair of the vertices can be assigned a weight. It was proven in [1] that, in the finite case, the question can be reduced to the situation when the Abelian group is a finite cyclic group. Nevertheless, in practical applications, the most common assignments of the weights are coming from the groups of numbers. For this reason, we consider the case when the underlying Abelian group G is torsion-free. We show that within the class of all torsion-free Abelian groups, the underlying transitive and unweighted graph predetermines the possibility of completion of a given transitive system. Such a graph is G-soluble. The question of G-solubility of a graph is equivalent to its ℝ-solubility. This, in turn, is equivalent to solvability of a certain system of linear equations in ℝ. As a result, G-solubility of a transitive graph can be determined by means of a feasible algorithm.

Definability of Completely Decomposable Torsion-Free Abelian Groups by Semigroups of Endomorphisms and Groups of Homomorphisms

Let C be an Abelian group. A class X of Abelian groups is called a CE• H-class if for any groups A, B ∈ X, it follows from the existence of isomorphisms E• (A) ≅ E• (B) and Hom(C,A) ≅ Hom(C,B) that there is an isomorphism A ≅ B. In this paper, conditions are studied under which the class \( {\mathfrak{I}}_{\mathrm{cd}}^{\mathrm{ad}} \) of completely decomposable almost divisible Abelian groups and class \( {\mathfrak{I}}_{\mathrm{cd}}^{\ast } \) of completely decomposable torsion-free Abelian groups A where Ω(A) contains only incomparable types are CE• H-classes, where C is a completely decomposable torsion-free Abelian group.

Quasi-Endomorphism Rings of Some Quasi-Decomposable Torsion-Free Abelian Groups of Rank 4

We obtain a description of quasi-endomorphism rings of torsion-free Abelian groups G of rank 4, quasi-decomposable into a direct sum of groups A1 and A2 of rank 1 and a strongly indecomposable group B of rank 2 in the case where the quasi-homomorphism group ℚ ⨂ Hom(A2,B) has rank 2.

Torsion-Free Abelian Groups of Finite Rank with Marked Bases

We define a one-to-one correspondence between torsion-free Abelian groups of finite rank with marked bases and finite sequences of elements of finitely presented modules over the ring of polyadic numbers. This correspondence is a duality of two categories.

On the Question of Definability of Homogeneously Decomposable Torsion-Free Abelian Groups by Their Homomorphism Groups and Endomorphism Rings

Let $$C$$ be an Abelian group. A class $$X$$ of Abelian groups is called a $$_CH $$ -class (a $$_CEH$$ -class) if, for any groups $$A$$ and $$B$$ in the class $$X$$ , the isomorphism of the groups $$\operatorname{Hom}(C,A)$$ and $$\operatorname{Hom}(C,B)$$ (the isomorphism of the endomorphism rings $$E(A)$$ and $$E(B)$$ and of the groups $$\operatorname{Hom}(C,A)$$ and $$\operatorname{Hom}(C,B)$$ ) implies the isomorphism of the groups $$A$$ and $$B$$ . In the paper, we study conditions that must be satisfied by a vector group $$C$$ for some class of homogeneously decomposable torsion-free Abelian groups to be a $$_CH$$ class (Theorem 1), and also, for some $$C$$ in the class of vector groups, for some class of homogeneously decomposable torsion-free Abelian groups to be a $$_CEH$$ -class (Theorem 2).

On the Question of Definability of Homogeneously Decomposable Torsion-Free Abelian Groups by Their Homomorphism Groups and Endomorphism Rings

Пусть $C$ - абелева группа. Класс $X$ абелевых групп назовем $_CH$-классом ($_CEH$-классом), если для любых групп $A$ и $B$ из класса $X$ из изоморфизма групп $\operatorname{Hom}(C,A)$ и $\operatorname{Hom}(C,B)$ (из изоморфизмов колец эндоморфизмов $E(A)$ и $E(B)$ и групп $\operatorname{Hom}(C,A)$ и $\operatorname{Hom}(C,B)$) следует изоморфизм групп $A$ и $B$. В статье исследуются условия, которым должна удовлетворять векторная группа $C$, чтобы некоторый класс однородно разложимых абелевых групп без кручения был $_CH$-классом (теорема 1), а также для некоторых $C$ из класса векторных групп, чтобы некоторый класс однородно разложимых абелевых групп без кручения был $_CEH$-классом (теорема 2) . Библиография: 15 названий.

Abelian RE-Groups

An Abelian group on which every nonzero ring is isomorphic to the ring of endomorphisms of this group is called an RE-group. In the present paper, the RE-groups are described in some classes of Abelian groups, including periodic, divisible, unreduced, and torsion-free rank-1 groups. It is shown that there are no RE-groups in the class of completely decomposable torsion-free Abelian groups.

Approximations for Theories of Abelian Groups

Approximations of syntactic and semantic objects play an important role in various fields of mathematics. They can create theories and structures in one given class by means of others, usually simpler. For instance, in certain situations, infinite objects can be approximated by finite or strongly minimal ones. Thus, complicated objects can be collected using simplified ones. Among these objects, Abelian groups, their first order theories, connections and dynamics are of interests. Theories of Abelian groups are characterized by Szmielew invariants leading to the study and descriptions of approximations in terms of these invariants. In the paper we apply a general approach for approximating theories to the class of theories of Abelian groups which characterizes the approximability of a theory of Abelian groups by a given family of theories of Abelian groups in terms of Szmielew invariants and their limits. We describe some forms of approximations for theories of Abelian groups. In particular, approximations of theories of Abelian groups by theories of finite ones are characterized. In addition, we describe approximations by quasi-cyclic and torsion-free Abelian groups and their combinations with respect to given families of prime numbers. Approximations and closures of families of theories with respect to standard Abelian groups for various sets of prime numbers are also described.

Definability of Completely Decomposable Torsion-Free Abelian Groups by Endomorphism Semigroups and Homomorphism Groups

Let C be an Abelian group. A class X of Abelian groups is called a CE•H-class if, for every groups A, B ∈ X, the isomorphisms E•(A) ≅ E•(B) and Hom(C, A) ≅ Hom(C, B) imply the isomorphism A ≅ B. In the paper, necessary and sufficient conditions on a completely decomposable torsion-free Abelian group C are described under which a given class of torsion-free Abelian groupsisa CE•H-class.

Definability of Completely Decomposable Torsion-Free Abelian Groups by Endomorphism Semigroups and Homomorphism Groups

Пусть $C$ - абелева группа. Класс $X$ абелевых групп назовем $ _CE^\bullet H$-классом, если для любых групп $A,B\in X$ из изоморфизмов $E^\bullet(A)\cong E^\bullet(B)$ и $\operatorname{Hom}(C,A)\cong\operatorname{Hom}(C,B)$ следует изоморфизм $A\cong B$. В статье описаны необходимые и достаточные условия на вполне разложимую абелеву группу $C$ без кручения, чтобы заданный класс абелевых групп без кручения был $ _CE^\bullet H$-классом. Библиография: 11 названий.

A Note on the Square Subgroups of Decomposable Torsion-Free Abelian Groups of Rank Three

AbstractA hypothesis stated in [16] is confirmed for the case of associative rings. The answers to some questions posed in the mentioned paper are also given. The square subgroup of a completely decomposable torsion-free abelian group is described (in both cases of associative and general rings). It is shown that for any such a group A, the quotient group modulo the square subgroup of A is a nil-group. Some results listed in [16] are generalized and corrected. Moreover, it is proved that for a given abelian group A, the square subgroup of A considered in the class of associative rings, is a characteristic subgroup of A.