It is well known that in a separable topological space every point-finite family of open subsets is countable. In the following we are going to show that both in σ-finite measure-spaces and in topological spaces satisfying the countable chain condition, point-finite families consisting of “large” subsets are countable.

Notation and terminology. Let A be a set. The family consisting of all (finite) subsets of A is denoted by . Let be a family of subsets of A. The sets and are denoted by and , respectively. We say that the family is point-finite (disjoint) if for each a ∈ A , the family has at most finitely many members (at most one member).

We shall prove the following about the “ringification” ρA of [2] and [5] of an archimedean l-group A: (a) Any “minimal ring” containing A is ρA; (b) A ↦ ρA is a reflector; (c) ρA need not be laterally complete when A is. These constitute the solutions to the problems posed in [2] by Paul Conrad.

1. The embedding into a ring. Let be the category which has objects archimedean l-groups A with distinguished positive weak unit eA, and morphisms l-group homomorphisms h: A → B with h(eA) = eB. Let be the category with objects archimedean f-rings R with identity 1R which is a weak unit, and morphisms l-ring homomorphisms h: R → S with h(lR) = 1S.

The distinct Hausdorff compactifications δX of a completely regular (Hausdorff) space X are in one-one correspondence with the admissible proximity relations δ on X, or alternatively, with the admissible totally bounded uniform structures for X. (See [1], [2].) Thus, δX is the Smirnov compactification of (X, δ). Generalized uniform structures for X will be described by means of pseudometrics on X (cf. [5], [7], [13]). Let where is in the proximity class π(δ) associated with (X, δ). Then a subset S of X is σ-discrete of gauge if , for all x, y ∈ S where x ≠ y.

All rings considered will be commutative with identity. By a graded ring we will mean a ring graded by the non-negative integers.

A ring R is called a π-ring if every principal ideal of R is a product of prime ideals. A π-ring without divisors of zero is called a π-domain. A graded ring (domain) is called a graded π-ring (-domain) if every homogeneous principal ideal is a product of homogenous prime ideals. A ring R is called a general ZPl-ring if every ideal is a product of primes. A graded ring is called a graded general ZPl-ring if every homogenous ideal is a product of homogeneous prime ideals.

In an earlier paper [14] I have adumbrated a method for establishing that the zeta-function of a Shimura variety associated to a quaternion algebra over a totally real field can be expressed as a product of L-functions associated to automorphic forms. Now I want to add some body to that sketch. The representation-theoretic and combinatorial aspects of the proof will be given in detail, but it will simply be assumed that the set of geometric points has the structure suggested in [13]. This is so at least when the algebra is totally indefinite, but it is proved by algebraic-geometric methods that are somewhat provisional in the context of Shimura varieties. However, contrary to the suggestion in [13] the general moduli problem has yet to be treated fully. There are unresolved difficulties, but they do not arise for the problem attached to a totally indefinite quaternion algebra, which is discussed in detail in [17].

1. Introduction. Let A be a sequence; throughout this paper sequences are understood to be infinite, strictly increasing and composed of non-negative integers. We define D, the infinite-difference set of A, to be the set of those non-negative integers which occur infinitely often as the difference of two terms of A. Plainly D has no positive terms if and only if ai+1 — ai → ∞ as i → ∞. Note that D contains zero. We shall be interested in the case when . Then D certainly contains more than one term. In fact, see Corollary 1, §2, in this case. Here and ḏ denote the (natural asymptotic) upper and lower density respectively.

1.1. We will denote by B the open unit ball in Cn, and we will denote by H(B) the class of all holomorphic functions on B. Let

Thus N(B) is convex (and compact in the compact open topology). We think that the structure of N(B) is of interest and importance. Thus we proved in [1] that if

(1.1)

if

(1.2)

and if n≧ 2, then g is an extreme point of N(B). We will denote by E(B) the class of all extreme points of N(B). If n = 1 and if (1.2) holds, then as is well known g ∈ E(B) if and only if

(1.3)

All spaces considered are completely regular. C*(X) denotes the set of bounded continuous real-valued functions in X. A subspace S of X is called C*-embedded in X if for every f ∈ C*(S) there is ϕ ∈ C*(X) with ϕ⥤S = f.

A space X is called almost compact if |βX – X| ≦ 1; basically disconnected if every cozero-set has open closure; extremally disconnected if every open set has open closure; an F-space if every cozero-set is C*-embedded; small if |C*(X)| = 2ω; and weakly Lindelöf if every open cover has a subfamily with and ⋃ dense. A point p of a space X is called a P-point of X if every Gδ-set in X which contains p is a neighborhood of p.

ω(X) denotes the weight of X.

Introduction. It is well-known that a group is uniquely determined by a system of generators, and a set of defining relations on those generators. Clearly it is of interest to find relations that are as simple as possible. In this paper, this question is dealt with for certain orthogonal groups of characteristic 2, which are generated by involutions.

Let V be a vector space over a field K of characteristic 2 (we always exclude the prime field K = G F (2)). Let Q be a quadratic form over V, and let S be the set of orthogonal transformations of (V, Q) whose path is 1-dimensional and not contained in the radical of V. Letting O* be the group generated by S, we shall show that every relation among generators in S is a consequence of relations of length 2, 3, or 4.

Let L(s, χ) be a Dirichlet L-function and let N(σ, T, χ) denote the number of zeros ρ of L(s, χ), counted according to multiplicity, in the rectangle σ ≦ Re(ρ) ≦ 1, |Im(ρ)| ≦ T, (T ≦ 1). In this paper we shall prove several new estimates for the sum

where Σ* denotes summation over primitive characters only. These estimates will all be of the type

(1)

where denotes any fixed positive quantity.

The geometry of Hjelmslev groups is a comprehensive plane metric geometry. It supplies, for example, an approach to euclidean, hyperbolic, elliptic, Minkowskian and Galilean geometry.

The subject of this geometry are Hjelmslev groups and their group planes. (The definition of Hjelmslev groups and some basic concepts and propositions of this theory can be found in the second edition of Bachmann's book [1] (pages 318-328). A similar report in English is the lecture [3]. A comprehensive introduction is developed in the key work [2]. The first part of this work was translated by Garner [4]. An abstract is given in Math. Rev. 52, 9066 (1976).) The group plane arises from giving geometric names to some group theoretical facts. The group plane of a Hjelmslev group is an incidence structure with orthogonality, and the Hjelmslev group acts on this plane as a group of motions.

In 1948, R. Frucht [5] proved that, given a finite group G, there are infinitely many connected cubic graphs X such that the automorphism group Aut X is isomorphic to G. In a letter, Professor Frucht has proposed the problem, whether in addition X can be required to be hamiltonian. One of the aims of the present note is to answer this question affirmatively.

THEOREM 1.1. Given a finite group G there are infinitely many finite hamiltonian cubic graphs Y such that Aut Y ≌ G.

In fact, we prove the following:

THEOREM 1.2. Given a finite cubic graph X having no component isomorphic to K4, there exists a hamiltonian cubic graph Y such that Aut Y ≌ Aut X and |V(Y)| = 6|V(X)|.

This implies 1.1 by the theorem of Frucht [5] mentioned above.

In Planar sublattices of a free lattice, I [8] we verify Jonsson's conjecture for finite planar lattices; in particular we obtain a characterization of finite planar sublattices of a free lattice among all finite lattices. In the present paper we use arguments of a quite different flavour to obtain another characterization. Let

be the family of lattices illustrated in Figures 1, 2, 3, and 4. Our goal is to prove the following theorem: a finite lattice is a planar sublattice of a free lattice if andonly if it does not have a member of as a sublattice.

1. Introduction. There are several multiplicative cohomology theories for which the group of units in the zeroth term is the zeroth term of another cohomology theory. Examples, due to Segal, May and others, are given by ordinary cohomology with rather general graded coefficients, real and complex K-theory with integral coefficients, and various bordism theories, also with integral coefficients [8, 7, 2, 5, IV]. The object of this paper is to show that complex K-theory modulo an odd prime p provides a counter-example.

To state the theorem precisely we recall the result of Araki and Toda that there is a unique anticommutative associative admissible multiplication in K*( ;Z/p) for p an odd prime [3, 3, 7, 10]; admissible is defined in [3] and means essentially that the reduction homomorphism K*( ) → K*( ; Z/p) preserves products.

This paper studies dualities (or contravariant category equivalences) between two categories of R-right and S-left modules which are finitely closed) that is, closed under submodules, factor modules and finite direct sums. Omitting the requirement that the categories contain all finitely generated modules from the classical Morita situation provides a generalization which substantially increases the number of such dualities.

We prove that a duality between two finitely closed categories A and B of modules is representable if and only if A and B consist of linearly compact modules. This encompasses work of Mueller ([7], [8]) for Morita dualities and of Goblot ([5], [6]).

All spaces in this paper are completely regular Hausdorff and all maps are continuous onto, unless otherwise stated. The purpose of this paper is to investigate the realcompactness of a space X which contains a Lindelöf space L such that every zero-set Z (in X) disjoint from L is realcompact. We show in § 2 that such a space X is very close to being realcompact (Theorems I, II and III). But in general such a space fails to be realcompact. Indeed, in §§ 3 and 4 the following questions of Mrówka [18, 19] are answered, both in the negative:

(Q. 1) If X = L ∪ G where L is Lindelöf closed and G is E-compact, then is X E-compact?

(Q. 2) Suppose f:X → Y is a perfect map such that the set M(f) = {y ∊ Y| |f−l(y)| > 1} of multiple points of f is Lindelöf (especially, countable) closed. If X is E-compact, is Y also E-compact?

The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and to the study of zeta-functions of Shimura varieties. In order to avoid disconcerting technical digressions I shall work with reductive groups over fields of characteristic zero, but the second assumption is only a matter of convenience, for the problems caused by inseparability are not serious.

The difficulties with which the trace formula confronts us are manifold. Most of them arise from the non-compactness of the quotient and will not concern us here. Others are primarily arithmetic and occur even when the quotient is compact. To see how they arise, we consider a typical problem.

F. B. Jones (1937) conjectured that every normal Moore space is metrizable. He also denned a particular kind of topological space (now known as Jones' spaces), proved that they were all non-metrizable Moore spaces, but was unable to decide whether or not Jones’ spaces are normal. J. H. Silver (1967) proved that a positive solution to Jones’ conjecture was not possible, and W. Fleissner (1973) obtained an alternative proof by showing that it is not possible to prove the non-normality of Jones’ spaces. These results left open the possibility of resolving the questions from the GCH. In this paper we show that if CH be assumed, then Jones’ spaces are not normal (Devlin, Shelah, independently) and that the GCH does not lead to a positive solution to the Jones conjecture (Shelah). A brief survey of the progress on the problem to date is also included.

The notion of L-indistinguishability, like many others current in the study of L-functions, has yet to be completely defined, but it is in our opinion important for the study of automorphic forms and of representations of algebraic groups. In this paper we study it for the simplest class of groups, basically forms of SL(2). Although the definition we use is applicable to very few groups, there is every reason to believe that the results will have general analogues [12].

The phenomena which the notion is intended to express have been met–and exploited–by others (Hecke [5] § 13, Shimura [17]). Their source seems to lie in the distinction between conjugacy and stable conjugacy. If F is a field, G a reductive algebraic group over F, and the algebraic closure of F then two elements of G(F) may be conjugate in without being conjugate in G(F).

For a non-principal Dirichlet character χ modulo q,

Let the Pólya-Vingradov inequality asserts that M(x) < q1/2 log q see [7]. in the opposite direction it is a trivial consequence of lemma 1 below and 1. Parseval's identity that if χ is primitive modulo q, then

We show that on average the latter of these estimates is the more precise.

THEOREM 1. For any real k > 0

where the summation is over all non-principal characters modulo q.

THEOREM 2. For any k > 0,