Chain conditions on subgroups of LCA groups

For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G,\(\widehat{G})\) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (H) of (G) that are easier to handle and show that \(\mathscr{C} (H)\) and \(\mathscr{C} (G/H)\) are large and embed, as a poset, in \(\mathscr{C}(G,τ)\). Important special results are: (i) if \(K\) is a compact subgroup of a locally quasi-convex group \(G\), then \(\mathscr{C}(G)\) and \(\mathscr{C}(G/K)\) are quasi-isomorphic (3.15); (ii) if (D) is a discrete abelian group of infinite rank, then \(\mathscr{C}(D)\) is quasi-isomorphic to the poset \(\mathfrak{F}_D\) of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group \(G \) with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset \( \mathscr{C} (G) \) is as big as the underlying topological structure of (G,τ) (and set theory) allows. For a metrizable connected compact group \(X\), the group of null sequences \(G=c_0(X)\) with the topology of uniform convergence is studied. We prove that \(\mathscr{C}(G)\) is quasi-isomorphic to \(\mathscr{P}(\mathbb{R})\) (6.9).

In [4], Edwin Hewitt denned a-rich LCA (i.e., locally compact abelian) groups and classified them by their algebraic structure. In this paper, we study LCA groups with some properties related to a-richness. We define an LCA group G to be power-rich if for every open neighbourhood V of the identity in G and for every integer n > 1, λ(nV) > 0, where nV = {nx ∈ G : x ∈ V} and λ is a Haar measure on G. G is power-meagre if for every integer n > 1, there is an open neighbourhood V of the identity, possibly depending on n, such that λ(nV) = 0. G is power-deficient if for every integer n > 1 and for every open neighbourhood V of the identity such that is compact, . G is dual power-rich if both G and Ĝ are power-rich. We define dual power-meagre and dual power-deficient groups similarly.

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

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.

The polynomial ring over a Goldie ring need not be a Goldie ring

A nearring N is weakly divisible (wd-nearring) if, for each x, y ∈ N, there exists an element z ∈ N such that xz = y or yz = x. In the first part the structure of wd-nearrings with ascending or descending chain conditions is studied and wd-nearrings with ascending chain condition on the right annihilators are completely described. In the second part all zerosymmetric wd-nearrings on the group G of integers (mod p n ), p prime, in which pG is the set of all the nilpotent elements are characterised and constructed. At the end the possibility to construct partially balanced block designs from finite wd-nearrings is investigated. Non balanced designs and their related association schemes are derived from wd-nearrings having cyclic additive group G of order 2 n . For wd-nearrings on a cyclic additive group of order p n , p prime and p ≠ 2, an example is given.

Extra invariance of shift-invariant spaces on LCA groups

In this paper, we determine the locally compact abelian (LCA) groups whose ring of continuous endomorphisms is local.

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.

A chain condition is a property, typically involving considerations of cardinality, of the family of open subsets of a topological space. (Sample questions: (a) How large a fmily of pairwise disjoint open sets does the space admit? (b) From an uncountable family of open sets, can one always extract an uncountable subfamily with the finite intersection property. This monograph, which is partly fresh research and partly expository (in the sense that the authors co-ordinate and unify disparate results obtained in several different countries over a period of several decades) is devoted to the systematic use of infinitary combinatorial methods in topology to obtain results concerning chain conditions. The combinatorial tools developed by P. Erdös and the Hungarian school, by Erdös and Rado in the 1960s and by the Soviet mathematician Shanin in the 1940s, are adequate to handle many natural questions concerning chain conditions in product spaces.

Let R be a commutative ring, with 𝔸(R) its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the annihilating-ideal graph of R, denoted by 𝔸𝔾(R). It is the (undirected) graph with vertices 𝔸(R)* ≔ 𝔸(R)\{(0)}, and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some finiteness conditions of 𝔸𝔾(R). For instance, it is shown that if R is not a domain, then 𝔸𝔾(R) has ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, Artinian). Moreover, the set of vertices of 𝔸𝔾(R) and the set of nonzero proper ideals of R have the same cardinality when R is either an Artinian or a decomposable ring. This yields for a ring R, 𝔸𝔾(R) has n vertices (n ≥ 1) if and only if R has only n nonzero proper ideals. Next, we study the connectivity of 𝔸𝔾(R). It is shown that 𝔸𝔾(R) is a connected graph and diam (𝔸𝔾)(R) ≤ 3 and if 𝔸𝔾(R) contains a cycle, then gr (𝔸𝔾(R)) ≤ 4. Also, rings R for which the graph 𝔸𝔾(R) is complete or star, are characterized, as well as rings R for which every vertex of 𝔸𝔾(R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

Chapter 7 continues to develop the theory of rings and studies chain conditions for ideals of a ring. The motivation came from an interesting property of the ring of integers Z that its every ascending chain of ideals terminates. This interesting property of Z was first recognized by the German mathematician Emmy Noether (1882–1935). This property leads to the concept of Noetherian rings. On the other hand, Emil Artin (1898–1962) showed that there are some rings in which every infinite descending chain of ideals terminates. Such rings are called Artinian rings. The ring Z has an infinite descending chain of ideals. This property of Z shows that Z is not an Artinian ring. In this chapter special classes of rings are also studied and deeper results on ideal theory are proved. Some finiteness condition for ideals is imposed and Noetherian rings are introduced which are versatile. The most convenient equivalent formulation of the Noetherian requirement is that the ideals of the ring satisfy the ascending chain condition. A connection between a Noetherian domain and a factorization domain is established. Moreover, the Hilbert Basis Theorem is proved. This theorem has made a revolution and has given an extensive class of Noetherian rings. Its application to algebraic geometry is also discussed. The study in this chapter culminates in rings with a descending chain condition for ideals, which determines their ideal structure.

In this paper, we introduce (Rees) artinian [Formula: see text]-acts as those acts that satisfy the descending chain condition on their (Rees) congruences. Then, we show that (Rees) artinian [Formula: see text]-acts are those acts whose factor acts are all finitely (Rees) cogenerated. In addition, we continue the study of (Rees) noetherian [Formula: see text]-acts, namely, those acts whose (Rees) congruences are finitely generated. Moreover, as a useful tool for the investigation of chain conditions, we prove that the properties of being (Rees) noetherian and artinian are inherited in Rees short exact sequences. Next, we specifically consider the chain conditions on monoids. Finally, we prove that every right Rees artinian, commutative monoid is right Rees noetherian.

Suppose that K = {K0, K1} is a partition of some finite combinatorial power [S]r of a set S. Let X(K) be the compact subspace of the Tychonoff cube [0, 1]S consisting of those functions ƒ whose supports Sƒ = {ƒ > 0} are 0-homogeneous i.e., [Sƒ]r ⊆ K0. It can be said that almost every example of a space witnessing the distinction between various chain conditions is a variation of X(K) (see [2]). This comes from the fact that the subject is closely related to the subject of Partition Calculus. (See [11] for an explanation of this point.) To have examples which are even more topologically interesting one usually tries to make them small, i.e., as closely related to the unit interval as possible. The operation X(K) has a considerable drawback in that respect. For example, if we want X(K) to be ccc this becomes equivalent to the fact that the poset of all finite 0-homogeneous sets is ccc which amounts to the fact that K1 must be very small. Hence K0 is big in the sense that there exist large 0-homogeneous sets. This usually results in X(K) having large size and having points of large character. One attempt to solve this problem was given by van Douwen in [5] by going to the subspace Xm(K) of X(K) consisting of those ƒ for which Sƒ are maximal 0-homogeneous subsets of S. Unfortunately, while Xm(K) usually does have small character it is almost never compact. This might have been the reason for his question ([2], p. 207) whether the Continuum Hypothesis implies that the class of all first countable compacta distinguishes the standard chain conditions that lie between ‘ccc’ and ‘separable’. In this paper we solve this problem completely. Moreover, we shall not go beyond the usual axioms of set theory in constructing the examples. The sequence of examples will start with a compact space of small character whose chain condition is not productive and it will end with a compact space of size c and small character which is ccc in a strong sense but which fails to have calibre θ for some regular uncountable cardinal θ, i.e., it fails to have the property of Shanin. Note that one cannot go further and show that, for example, compact spaces of small character distinguish between ‘the property of Shanin’ and ‘separable’. This follows from an old result of Efimov [3] that, under CH, first-countable spaces of calibre ℵ1 are separable. The combinatorics behind our examples have been developed in a series of papers that deal with the subject of forcing axioms in general and Martin's axiom in particular ([10, 11, 12, 13, 14]). Martin's axiom was originally invented in connection with the Souslin Problem, i.e., to show that certain compact ccc spaces must be separable (see [8]). A result of the aforementioned study of MA showed that this axiom is nothing more than the statement that all compact ccc spaces of π-weight < c must be separable (see [12]). An analysis of the fact that MA implies SH, due to Hajnal and Juhasz[6] (see also [4], §43 for a definite result in that direction due to Shapirovskii), revealed that MA implies that every compact ccc space X with the property χ(X)+ < c must be separable. This result explains why the compact ccc non-separable spaces X that we construct in this paper have the property that χ(x, X) < c for all x m X rather than the stronger property χ(X) < c or even χ(X) = ℵ0. Note that our examples show, answering a question from [6], that the Hajnal–Juhasz result is sharp in the sense that the assumption χ(X)+ < c cannot be weakened to χ(X) < c. The first example to show this was constructed by Bell [1] using a consequence of MA rather than just ZFC for its construction. Another feature of our examples is that they all are remainders of certain compactifications of the integers. This is of independent interest in certain constructions of weak P-points in compact F-spaces. An explanation of this can be found in [1] and [9] where the first examples of ccc non-separable remainders were constructed and used.

We obtain a chain condition for dividing in an arbitrary theory and a new and shorter proof of a chain condition result of Shelah for simple theories.