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.

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)

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.

Let (X, ) be a topological space equipped with a partial order ≦ and let C (≦) denote the continuous increasing functions mapping X into R (a function f : X → R is increasing provided f(x) ≦ f(y) whenever x ≦ y) Then (X,, ≦) is an N-space (in the terminology of [16], a completely regular order space) provided is the weak topology of C (≦) and if x ≦ y is false, then there is an f ∈ C (≦) such that f(y) < f(x). L. Nachbin's introduction of N-spaces was perspicacious, for these spaces now find application in a wide spectrum of mathematics.

An equidistant permutation array (EPA) A(r λ v) is a v × r array defined on a set V of r symbols such that every row is a permutation of V and any two distinct rows have precisely λ common column entries. Define R(r, λ) to be the largest value of v for which there exists an A (r, λ; v). Deza [2] has shown that

where n = r – λ. Bolton [1] has shown that

(*)

In this paper, we show that equality holds in (*) for λ > ┌n/3┐(n2 + n). In order to do this we require several more definitions.

Generalized axisymmetric potentials Fα (GASP) are regular solutions to the generalized axisymmetric potential equation

(1.1)

in some neighborhood Ω of the origin where they are subject to the initial data

(1.2)

along the singular line y = 0. In Ω, these potentials may be uniquely expanded in terms of the complete set of normalized ultraspherical polynomials

(1.3)

defined from the symmetric Jacobi polynomials Pn(α, α)(ξ) of degree n with parameter α as Fourier series

(1.4)

Graphs which are 2-connected have been called blocks in the graph theoretic literature [3] and stars in the parlance of statistical mechanics [1]. They are also called nonseparable graphs, and in this paper include the complete graph K2 on p = 2 points. The standard terminology of graphical enumeration can be found in [3].

When its points are colored using two distinct colors in such a way that adjacent points receive different colors, a graph is called 2-colored. If the colors are considered interchangeable, such a graph is called bicolored. A graph which can be bicolored is called bicolorable. It is obvious that a connected bicolorable graph has a unique bicoloring, and so one can speak of the sizes of its color classes.

In [8], McKenzie introduced the notion of a bounded homomorphism between lattices, and, using this concept, proved several deep results in lattice theory. Some of these results were intimately connected with the work of Jónsson and Kiefer in [6] where an attempt was made to characterize finite sublattices of free lattices. McKenzie's characterization and others that followed (see [7] and [5]) still have not answered the (now) celebrated Jônsson conjecture:

A finite lattice is a sublattice of a free lattice if and only if it satisfies (SD∨), (SD∨) and (W).

(The properties mentioned here are defined in the text.)

Let S denote the unit sphere in Cn, B the (open) unit ball in Cn and H∞(B) the collection of all bounded holomorphic functions on B. For f ∈ H∞(B) the limits

exist for almost every ζ in S, and the map ƒ → ƒ* defines an isometric isomorphism from H∞(B) onto a closed subalgebra of L∞(S), denoted H∞(S). (The only measure on S we will refer to in this paper is the Lebesgue measure, dσ, generated by Euclidean surface area.) Rudin has shown in [4] that the spaces H∞(B) + C(B) and H∞(S) + C(S) are Banach algebras in the sup norm. In this paper we will show that the maximal ideal space of H∞(B) + C(B), Σ (H∞(B) + C(B)), is naturally homeomorphic to Σ (H∞(B)) and that Z (H∞(S) + C(S)) is naturally homeomorphic to Σ (H∞(S))\B.

SL(2,C) is the group of all complex unimodular 2 × 2 matrices. A subgroup of SL(2, C) is said to be discrete if it does not contain any convergent sequence of distinct elements. A subgroup is said to be elementary if the commutator of any two elements of infinite order has trace 2. The discreteness condition which this note relates to is the following:

PROPOSITION 1. If two complex, unimodular 2 × 2 matrices X and Y generatea non-elementary, discrete group, then

A ring R (with identity element) is called a right PCI-ring if and only if every proper cyclic right R-module is injective; that is, if C is a cyclic right R-module then either C ≌ R or C is injective. Faith [3, Theorems 14 and 17] (or see [2, Proposition 6.12 and Theorem 6.17]) proved that if a ring R is a right PCI-ring then R is semiprime Artinian or R is a simple right semihereditary right Ore domain. These latter rings we shall call simple rightPCI-domains. Examples of non-Artinian simple right PCI-domains were produced by Cozzens [1]. The object of this paper is to examine rings with similar properties and thus extend Faith's results.

Coinitial graphs were used in [2; 3 ; 4] as a combinatorial tool in the Reidemeister- Schreier process in order to prove subgroup theorems for Fuchsian groups. Whitehead had previously introduced such graphs but used topological methods for his proofs [8; 9]. Subsequently Rapaport [7] and Iliggins and Lyndon [1] gave algebraic proofs of the results in [9], and AIcCool [5; 6] has further developed these methods so that presentations of automorphism groups could be found.

In this paper it is shown that Whitehead automorphisms can be described by a “cutting and pasting” operation on coinitial graphs. Section 1 defines and gives some combinatorial properties of these operations, based on [1].

Etant donné une variété différaitiable M, deux pseudogroupes ℒ1 et ℒ2 sur M sont dits appartenir au même type, s'il existe un difféomorphisme local de M transformant ℒ1 en ℒ1

Le problème général de la recherche des pseudogroupes sur M peut être énoncé de manière suivante:

(1) Déterminer tous les types des pseudogroupes sur M, et pour chaque type un représentant particulier.

(2) Pour chaque pseudogroupe ainsi obtenu, déterminer le plus grand pseudogroupe sur M dans lequel il est invariant.

Dans [5], E. Cartan a utilisé cette méthode pour classifier les pseudogroupes sur R2; pour une démonstration plus récente voir [6]. Ceci montre ce plus l'intérêt d'étudier le normalisateur d'un pseudogroupe de Lie ℒ sur M.

A continuum will be a connected, compact, metric space. By a mapping we mean a continuous function. By a partially ordered space X we mean a continuum X together with a partial order which is closed when regarded as a subset of X × X. We let 2x (resp. C(X)) denote the hyperspace of closed subsets (resp. subcontinua) of X with the Vietoris topology which coincides with the topology induced by the Hausdorff metric. The hyperspaces 2X and C(X) are arcwise connected metric continua (see [3, Theorem 2.7]). If A ⊂ X we let C(A) denote the subspace of subcontinua of X which lie in A.

If X is a partially ordered space we define two functions L, M : X → 2X by setting for each x ∊ X

The question under what conditions a closed invariant subspace possesses a closed invariant complement is of major importance in operator theory. In general it remains unanswered. In this paper we drop the requirement that the invariant complement be closed. We show in section 1 that the question is answerable under fairly mild conditions for a quasinilpotent operator (Theorem 1.5). These conditions will cover the case of a quasinilpotent operator with dense range and no point spectrum. In section 2 we discuss the consequences for the Volterra operator V. Since V is unicellular, its proper closed invariant subspaces do not possess closed invariant complements. However, they are all algebraically complemented (Proposition 2.1).

Let S be a subnormal operator on a Hilbert space ℋ and let N be its minimal normal extension on the Hilbert space ℋ. (We refer the reader to [5, 15] for the basic material on subnormal operators.) Denote the commutant and double commutant of an operator T by ﹛T﹜’ and ﹛T﹜”, respectively.

A semisimple symmetric space can be defined as a homogeneous space G/H, where G is a semisimple Lie group, H an open subgroup of the fixed point group of an involutive automorphism of G. These spaces can also be characterized as the affine symmetric spaces or pseudo-Riemannian symmetric spaces or symmetric spaces in the sense of Loos [4] with semisimple automorphism groups [3, 4]. The connected semisimple symmetric spaces are all known: they have been classified by Berger [2] on the basis of Cartan's classification of the Riemannian symmetric spaces. However, the list of these spaces is much too long to make a detailed case by case study feasible. In order to do analysis on semisimple symmetric spaces, for example, one needs a general structure theory, just as in the case of Riemannian symmetric spaces and semisimple Lie groups.

The class S of functions under study in this paper was introduced by V. I. Smirnov in 1932. This class was subsequently investigated by various authors, a pertinent paper to the present wrork being that of Tumarkin and Havinson [2], who showed that a plane compact set of logarithmic capacity zero is 5-removable. Another important development, due to Yamashita [3], wras that the class 5 could be characterized as those analytic functions ƒ for which log+ |ƒ| has a quasi-bounded harmonic majorant.

In what follows, we discuss the Smirnov class in the context of planar surfaces, exploiting some ideas in the work of Hejhal [1] to establish that a closed, bounded, totally disconnected set is S-removable if and only if its complement belongs to the null class Os.

In [2, Theorem 3.2, p. 429] and [3, Theorem 2.1, p. 155], the author established two induction theorems which gave rise to a series of fundamental results in spectral and rearrangement inequalities. In particular, the classical inequalities of Hardy-Littlewood-Pólya [4, Theorem 108, p. 89] and Pólya [6] were derived and conditions for equalities obtained (see [2, Theorem 3.8, p. 433] and [3, Theorem 2.6, p. 157]). In [5, Theorem 6, p. 651 ; and Theorem 20, p. 569], Fischer and Holbrook also gave alternative conditions for equalities to hold in the aforesaid inequalities. In this paper, we show that the result of Fischer and Holbrook can be proved by induction using an inductive rearrangement theorem which turns out to be a stronger version of the induction theorems given in [2, Theorem 3.2, p. 429] and [3, Theorem 2.1, p. 155].

Let A be a ring (not necessarily associative) in which 2x = a has a unique solution for each a ∈ A. Then it is known that if A contains an identity element 1 and an involution j : x ↦ x and if Ja is the canonical involution on An determined by

where the ai al−l, 1 ≦ i ≦ n are symmetric elements in the nucleus of A then H(An, Ja), the set of symmetric elements of An, for n ≧ 3 is a Jordan ring if and only if either A is associative or n = 3 and A is an alternative ring whose symmetric elements lie in its nucleus [2, p. 127].