Set Theory and Structure Theorems

In the paper, necessary and sufficient conditions for an Abelian group A to be isomorphic to the endomorphism group End(A) are obtained. The classes of periodic Abelian groups, divisible Abelian groups, nonreduced Abelian groups, and reduced algebraically compact Abelian groups are considered. For certain classes of Abelian groups, the isomorphism problem for a group and its endomorphism group is solved under the assumption that the endomorphism group itself has the corresponding property.

The fundamental theorem of abelian groups states that any finitely generated abelian group is a direct sum of cyclic groups. This theorem plays a fundamental role in the structure theory of abelian groups. It has fascinated many algebraists to look at this theorem from different points of view:- (a) To find suitable generalizations of this theorem for modules over certain classes of rings, e.g. Dedekind domain, hereditary noetherian prime rings, valuation rings etc. (b) To create suitable versions of this theorem in some module categories and use such versions to develop the structure theory of such module categories. For example, torsion abelian group-like modules, modules with finitely generated submodules direct sums of multiplication modules. (c) To find those rings for which certain versions of the fundamental theorem of abelian groups hold. For example rings over which all finitely generated modules are direct sums of cyclic modules. (d) To examine the structure of certain classes of abelian groups and to try to find modules with similar structures. (e) To find roles of answers to some of above types of questions in the general theory of modules.

Let $$\Lambda$$ be a class of Abelian groups. A group $$A\in\Lambda$$ is said to be determined by its endomorphism semigroup $$E^\star(A)$$ in the class $$\Lambda$$ if every isomorphism $$E^\star(A)\cong E^\star(B)$$ , where $$B\in\Lambda$$ , implies the isomorphism $$A\cong B$$ . The paper describes those Abelian groups in the class $$\mathscr Q\mathscr D_{\mathrm{cd}}$$ of completely decomposable quotient divisible Abelian groups which are determined by their endomorphism semigroups in the class $$\mathscr Q\mathscr D_{\mathrm{cd}}$$ .

The classical structure theorems of algebraic systems usually assume some type of finiteness condition. The most common finiteness restriction is a chain condition. Thus the proofs of the fundamental structure and decomposition theorems for lattices have customarily required the ascending chain condition. Moreover, these theorems generally fail to hold in arbitrary lattices. Nevertheless, there are important examples of decomposition theorems for lattices associated with abelian groups and rings in which the ascending chain condition does not hold. These lattices have in common another distinctive property—they are compactly generated. Namely, the lattice is generated by a collection of elements which are finitely dependent in the sense that any such element is contained in the union of a set of lattice elements if and only if it is contained in the union of a finite subset. The compact elements of the lattice of ideals of a ring are the finitely generated ideals. Likewise the compact elements of a lattice of congruence relations are the congruence relations generated by collapsing a finite collection of element pairs. More generally, the lattice of congruence relations and the lattice of subsystems of a universal algebra are compactly generated. Since structure theorems for an algebraic system correspond to decomposition theorems in the lattice of congruence relations, this strongly suggests that compactly generated lattices are the appropriate domain in which to study decomposition theory. Furthermore, since every lattice satisfying the ascending chain condition is trivially compactly generated, it follows that the classical case will be subsumed in the more general theory.

It is proved that all cohomology groups of a group G acting on an abelian group of a certain type vanish if the action of some subnormal abelian subgroup of G is fixed-point-free.This result is then applied to obtain results about group extensions and about the complete reducibility of linear groups.Introduction.The purpose of this paper is to prove the Proposition below and its corollary, Theorem 1, and to derive from the latter some consequences about group extensions and about the complete reducibility of linear groups.These results are contained in Corollaries 2-4.Statement (a) of § 1 and part (b) of the Lemma, §2, are related to the surrounding material, but are not used in the subsequent proofs.I am indebted to several colleagues for useful suggestions in connection with this paper.I am also grateful to the referee for his valuable comments, especially for pointing out that the original version of Theorem 1 could be strengthened to its present form.In what follows, a Vx-module for a group G will mean a finite direct sum 2 V¡, where V¡ is a finite-dimensional vector space over a field k¡, and V¡ is a /c,G-module.Fory > 2, a VJ-module is a G-module A for which there is a short exact sequence 0 -> A' -> A -» A" -» 0, where A' is a K'-module and A" is a F-7-1-module.A is a W-module if it is a F^-module for somtj.It can be shown that every iV-module is a direct sum of a torsion group of finite exponent and a torsion free divisible group (i.e., a vector space over Q).Conversely, one can show that every such abelian group is a W-mod\i\c (for trivial group action).Moreover, if such a group A = T © F is given as a Gmodule, then A is automatically a W-module for G if the torsion part T is finite and Fis finite dimensional over Q.This follows by induction on \T\, the order of T, from the exact sequence 0 -> Sp -* A -> A/Sp = T/Sp © F '-* 0, wherep is any prime dividing |7j and Sp = [x G A: px = 0).(If T = 0, the statement is clear.)Proposition.Let G be a group, N a subnormal abelian subgroup of G, k a

A class of Abelian groups with hereditary rings of endomorphisms

For some classes of Abelian groups, an answer to the following question is presented: when is the union of homomorphic images of a group a subgroup of another group? In connection with this, the concept of a homomorphically stable group is introduced, and homomorphic stability of groups from different classes of Abelian groups is studied.

If H is a subgroup of an abelian group G and φ ∈ End(G), H is called φ-inert (and φ is H-inertial) if φ(H) ∩ H has finite index in the image φ(H). The notion of φ-inert subgroup arose and was investigated in a relevant way in the study of the so called intrinsic entropy of an endomorphism φ, while inertial endo-morphisms (these are endomorphisms that are H-inertial for every subgroup H) were intensively studied by Rinauro and the first named author.A subgroup H of an abelian group G is said to be fully inert if it is φ-inert for every φ ∈ End(G). This property, inspired by the “dual” notion of inertial endomorphism, has been deeply investigated for many different types of groups G. It has been proved that in some cases all fully inert subgroups of an abelian group G are commensurable with a fully invariant subgroup of G (e.g., when G is free or a direct sum of cyclic p-groups). One can strengthen the notion of fully inert subgroup by defining H to be uniformly fully inert if there exists a positive integer n such that |(H + φH)/H| ≤ n for every φ ∈ End(G). The aim of this paper is to study the uniformly fully inert subgroups of abelian groups. A natural question arising in this investigation is whether such a subgroup is commensurable with a fully invariant subgroup. This paper provides a positive answer to this question for groups belonging to several classes of abelian groups.

We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a dense subset. (One shows first that the dense is order isomorphic to the rationals, Q, and then that the ordered set is isomorphic to the Dedekind completion of its dense subset.) Souslin raised the question as to whether the countable dense subset condition could be replaced by the following consequence [15]: (*) Every disjoint family of non-empty open intervals is countable.' We use SH (Souslin's Hypothesis) to denote the following proposition: Every order complete order dense linearly ordered set satisfying ( * ) contains a dense subset. We use ZFC to denote Zermelo-Fraenkel set theory (including the axiom of choice). In [16], Tennenbaum constructed models of ZFC in which SH is false. Moreover, in one of these models the continuum hypothesis (CH) is false, while in another one, the generalized continuum hypothesis (GCH) is true. Thus SH is independent of the usual axioms of set theory. (This result is due, independently, to Jech [7].)2

Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of Ext\ for rings A of global dimension 1.Using Martin's Axiom, the necessity of these conditions is shown not to be a theorem of ZFC.Applications are given to abelian group theory, including a partial solution (assuming V = L) to a problem of Baer on the splitting of abelian groups.Some independence results in abelian group theory are also proved.Introduction.In a remarkable recent paper [22], S. Shelah proved that Whitehead's problem is unsolvable in ordinary set theory (Zermelo-Frankel set theory with the Axiom of Choice).More precisely, he proved that two different answers to the problem are obtained when set theory is extended by adding two different axioms: the Axiom of Constructibility and Martin's Axiom.(See Theorem 4.1.)Shelah's arguments were specific to Whitehead's problem and nonhomological in character.In this paper we extend Shelah's techniques in order to obtain some general results about the vanishing of ExtA (for modules over a ring A of global dimension 1) in different models of set theory, and we apply them to some specific problems in abelian group theory.Assuming the Axiom of Constructibility we obtain necessary and sufficient conditions for the vanishing of ExtA (Theorems 1.2 and 1.5).To state the result in.a special case: if C and A are abelian groups of cardinality N,, Ext(C, A) -0 if and only if C is the union of an increasing chain of countable subgroups [Cv\v < «,} such that the chain is smooth (i.e., C0 -U"<oC, for aii iimit ordinals a) and satisfies Ext(C0, A) = 0 and E\i(Cv+x/Cv,A) = 0 for all v < K,.Although the sufficiency of this condition is an easy consequence of classical homological algebra (see Theorem 1.2), the proof of necessity makes use of the Axiom of Constructibility (Theorem 1.5).The necessity of this condition is not provable in ordinary set theory; in particular the condition is not necessary when Martin's Axiom is assumed ( §3).

This paper gives an informal introduction to structure theory for minor- closed classes of matroids representable over a fixed finite field. The early sections describe some historical results that give evidence that well-defined structure exists for members of such classes. In later sections we describe the fundamental classes and other features that necessarily appear in structure theory for minor-closed classes of matroids. We conclude with an informal statement of the structure theorem itself. This theorem generalises the Graph Minors Structure Theorem of Robertson and Seymour.

This article presents a formal description of object models in terms of a unified object-oriented metamodel for database applications, developed by the author and independent of the domain. Described metamodel is implemented by the author in own development environment which called SharpArchitect RAD Studio. In the beginning of the article the available works of similar subjects are thoroughly analyzed. Analysis have revealed that most authors use the set theory and predicate calculus for representation of the elements of object-oriented programming languages and design of object models. The same approach is used in this article. This article presents a metamodel, developed by the author for the design and implementation for object-oriented database applications. Attention is paid to the hierarchy of the classes presented in the application domains. Also the composition and structure of classes that allows to present both atomic literal types and classes of attributes which values are the instances of classes and collection of copies are described in detail. Attention is paid to the implementation of the dynamic component by description of methods (with support of the parameters) and present classes events. The formal approach to the description of the models for any application domains are proposed on the basis of the metamodel with using of set theory. The unified model of testing of objectoriented applications development tools was developed for the analysis of the applicability of the proposed approach. Optimality criteria of this model as well as its implementation in C# with the representation of all main classes presented in it are described. The proposed formal approach has been tested on the described model. Domain classes, attributes, associations and whole hierarchies have been described. It allowed to confirm the validity of the proposed formal solution. In conclusion the assumptions about the directions of further development of the proposed approach are made.

In the algebra there are several kinds of structure theorems which can be formulated without operations, using only homological tools. For instance, the wellknown fact that any universal algebra can be subdirectly embedded in a direct product of subdirectly irreducible algebras, can be formulated in a pure categorytheoretical manner. Now the question arises what its dual statement asserts. Our purpose is to give such a category which satisfies certain selfdual conditions, and making use of these, to prove structure theorems and their dual statements. The structure theorems themselves are, of course, well-known statements for algebraic stuctures. However, their duals yield some theorems of unusual type. About the possibility of the dualization there occurs some trouble. The most of the difficulties is at finding selfdual conditions being necessary to prove the theorems. So we must not make use of the condition 'every epimorphism is a normal one' which is fulfilled for groups, since its dual is false. Further the lattice of all congruencerelations of any universal algebra is a so-called compactly generated lattice. This fact plays a very important role in the proof of the theorem according to subdirect embeddings of universal algebras, nevertheless compactly generating is not a se!fdual notion. Applying the theorems proved for certain categories, we establish some particular theorems for rings, groups, modules, respectively. In w 2 we give a detailed enumeration of the usual notions and assertions of the theory of categories with respect to the importance of the dual notions and assertions, moreover, we form a system of selfdual conditions which will be satisfied by the category we are dealing with. w 3 is devoted to the investigation of subdirect embeddings, subdirect irreducibility and to the dualization of those. In w 4 we applicate the results developed before for rings, groups, module ~ and abelian groups. Most of the applications are concerned with rings.

Monitoring secondary forest regrowth is a priority in forest restoration strategies. A site history helps in understanding the present status of natural regeneration in the three landscapes impacted by bauxite mining. Nonetheless, high rainfall in bauxite residue storage areas can facilitate natural regeneration of forest. This research analyzed the natural regeneration of forest after thirty years of different land use histories at three bauxite mining areas of the Upper Demerara—Berbice region of Guyana. There are no man made forest plantations in the three landscapes being reviewed. The methodology included: 1) the selection of three sampling landscapes with different land use histories 2) the generation Land Use/Land Cover maps using KMeans unsupervised classification of satellite images in each landscape and 3) the assessment of landscape structure of the land cover classes for year 2020 at class and landscape level using landscape metrics. The assessment of landscape structure of the land cover classes was carried-out with landscape metrics for the comparisons at class and landscape level. Principal component analysis enables the identification of main patterns among landscape-level metrics and land cover classes. Discriminant classification of the landscape classes was analyzed with the different metrics. The results suggest that Normalized Difference Vegetation Index and KMeans unsupervised classification can be used to evaluate the difference in natural forest regeneration among landscapes with differing land use histories. The landscape metrics revealed secondary stages of forest succession. The Landscape Shape Index and Edge Density were the most significant for landscape differentiation. The result of the various land uses reveals a mosaic of early, intermediate, and late successional sequences.

Radicals in Abelian Groups