Abelian groups with partial decomposition bases in 𝐿_{∞𝜔}^{𝛿}, Part I

§1. Introduction.The motivation of this work comes from two different directions: infinite abelian groups, and ordered algebraic structures. A challenging problem in both cases is that of classification. In the first case, it is known for example (cf. [KA]) that the classification of abelian torsion groups amounts to that of reducedp-groups by numerical invariants called theUlm invariants(given by Ulm in [U]). Ulm's theorem was later generalized by P. Hill to the class of totally projective groups. As to the second case, let us consider for instance the class of divisible ordered abelian groups. These may be viewed as ordered ℚ-vector spaces. Their theory being unstable, we cannot hope to classify them by numerical invariants. On the other hand, being o-minimal, the theory enjoys several good model theoretic properties (cf. [P-S]), so the search for some reasonable invariants is well motivated. The common denominator of the two cases, as well as of many others, is valuation theory. Indeed given an ordered vector space, one can consider it as a valued vector space, endowed with the natural valuation. Also, the socleG[p] of a reduced abelianp-groupG, endowed with the height functionhG, is a valued vector space over(the prime field of characteristicp) with values in the ordinals (cf. [F]).

Let H be a subgroup of the free abelian group G.In order for there to exist a basis {*,},6/ of G for which H = ieI(nXj)for suitable nonnegative integers n, it is obviously necessary for G/H to be a direct sum of cyclic groups.In the 1950's, Kaplansky raised the question of whether this condition on G/H is sufficient for the existence of such a basis.J. Cohen and H. Gluck demonstrated in 1970 that the answer is "yes"; their result is known as the stacked bases theorem, and it extends the classical and well-known invariant factor theorem for finitely generated abelian groups.In this paper, we develop a theory that contains and, in fact, generalizes in several directions the stacked bases theorem.Our work includes a complete classification, using numerical invariants, of the various free resolutions of any abelian group.

Classification of group gradings on simple Lie algebras of types [formula omitted], [formula omitted], [formula omitted] and [formula omitted

In [8], odd dimensional knot cobordism groups C2n^ (n'2>2) were studied and given a purely algebraic description. C2 Λ_j is the set of all cobordism classes of knotted (2n— l)-spheres in the (2n+l)-sphere, and it is an abelian group under connected sum. In this paper we extend this to knotted homology 3-spheres in the 5-sphere and consider their homology cobordism classes. The set of all such homology cobordism classes, denoted by Cy is also an abelian group under connected sum. We shall describe the group Cf using the usual homology cobordism group of homology 3-spheres and Levine's cobordism group of certain matrices (Theorem 1.1). Our argument heavily depends on that of J. Levine. In §3 we introduce the bounding genus and the Murasugi number of a knotted homology 3-sphere in S*. Using our techniques we can estimate these numerical invariants from above. In §4 we consider the case of algebraic 3knots which appear around isolated singularities of complex hypersurfaces in C (see [11]). For example, we shall show that two algebraic 3-knots which are homology 3-spheres and defined by weighted homogeneous polynomials are homology cobordant if and only if they are isotopic. Throughout the paper, we shall work in the C°° category. Homology groups will always be with integral coefficients. The author would like to thank Professor S. Fukuhara for suggesting the problem. He is also grateful to Professor F. Michel who was kind enough to send us a copy of S. Akbulut's note.

The concept of physical and fractal dimension I. The projective dimensions

We give a complete description of the topological spaces X such that the free abelian topological group A(X) embeds into the free abelian topological group A(I) on the closed unit interval. In particular, the free abelian topological group A(X) on any finite-dimensional compact metrizable space X embeds into A(I). To obtain our description, we study similar embeddings of the free locally convex spaces and continuous surjections between the spaces of continuous functions with the pointwise topology. Proofs are based on the classical Kolmogorov's Superposition Theorem.

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.

Introduction. In [7] and [8] we have determined the isomorphisms of the endomorphism rings of two special classes of torsion-free modules. In [7] the modules were free, over principal left ideal domains, and the isomorpnism was always induced by a semi-linear transformation of the modules. In [8] the ring of scalars was a complete discrete valuation ring, but no further restriction was placed on the modules. Except for the special case when both modules are divisible, the same result holds as in [7]. In each case a key point in the argument is that a certain indecomposable direct summand can be shown to be cyclic. Even for ordinary torsion-free abelian groups, the abundance of the indecomposable ones makes it impossible to prove that isomorphism of the endomorphism rings implies isomorphism of the underlying groups. We show first that for any cardinal number c, there exist homogeneous completely decomposable groups G and H of rank c which are not isomorphic, but have isomorphic rings of endomorphisms. This makes it clear that strong restrictions must be placed on the class of groups considered if it is hoped that the ring isomorphism is to be induced by a group isomorphism. In particular, homogeneity coupled with separability or complete decomposability is not sufficient. We are, however, able to generalize the results of [7] to a class of what we call locally free modules A over principal left ideal domains F. A locally free abelian group (F is the ring of integers) is just a homogeneous separable group of null type [6, p. 208]. This latter class of groups includes the complete direct sum of infinite cyclic groups, as well as all its subgroups [4, p. 176]. For example, the set of all bounded sequences of integers is such a group. We also determine (Theorem B) which semi-linear transformations induce the same ring isomorphisms. The results are similar to the vector space case considered in [2], and apply to the endomorphism rings of locally free modules as well as the torsion-free modules of [8].

The results of Szele and Szendrei [‘On Abelian groups with commutative endomorphism rings’, Acta Math. Acad. Sci. Hungar.2 (1951), 309–324] characterizing abelian groups with commutative endomorphism rings are generalized to modules whose endomorphism rings have various restrictions on their idempotents. Such properties include central or commuting idempotents, and one-sided ideals being two-sided. Related properties include direct summands having unique complements, or being fully invariant.

The canonical mapping on the product of a LCA group with its dual is shown to extend uniquely in a homomorphic and analytic way to the product of the respective complexifications.

Endoregular modules

Permutation Summands Over Z

Properties of abelian groups related to a given finite rank torsion free abelian group A are analyzed in terms of End (A), the endomorphism ring of A. This point of view gives rise to generalizations of some classical theorems by R. Baer and examples of pathological direct sum decompositions of finite rank torsion free abelian groups.

Matchings in arbitrary groups

On the Schröder–Bernstein property for modules