Recent efforts to generalize a classic result of Hajos [3] on the decomposition of finite abelian groups into direct sums of subsets (see Fuchs [1, Chap. XV]) led B. Gordon [2] to the following conjecture. If are r-dimensional row vectors over GF(3) such that: (i) Any weighted (±) sum of any even number of 's is nonzero, (ii) For each r-dimensional , there exists an s such that

Then there exists a subset of either 1 or 4 's which satisfies the same conditions.

This paper proves Gordon's conjecture.

A lower bound for a family of Ramsey numbers is derived using a geometrical argument.

A ring R with identity element is n-finite if for any pair A, B of n × n matrices over R, AB = In implies BA = In. In module theoretic terms, R is n-finite if and only if in a free R-module of rank n any generating set of n elements is free. If R is n-finite for all positive integers n then R is said to be strongly finite. It is known that all commutative rings, all Artinian rings and all Noetherian rings are strongly finite. These and many other interesting results appear in a paper of P. M. Cohn [1].

The main aim of this paper is to study the convergence of a sequence of contractions in a generalized metric space. More specifically, we investigate the following question:

"If a sequence of contractions {fr} with fixed points ur (r = 1,2,...) converges to a mapping f with a fixed point u, under what conditions will the sequence ur converge to u?"

A partial answer to the above question has been given in metric spaces by Bonsall [1]. This result has since been improved by Russell and Singh [6]. Further results will now be given in a generalized metric space.

Although a nontopological concept, boundedness seems to be of considerable importance in a topological space. 'There are many topological problems in which it is essential to be able to make this distinction' (between bounded and unbounded sets) [1]. Boundedness and in particular boundedness-preserving' uniform spaces appear to have applications to topological dynamics [4].

In spite of this importance, there have been only isolated attempts at developing the concept. Alexander [1] and Hu [7] tried the axiomatic approach. Hu, for example, calls a nonempty family of sets a boundedness if is hereditary and closed under finite union.

Let R be a ring with 1. All modules considered are to be unital left R-modules unless otherwise noted.

Definition. A σ-set for R is a nonempty set Σ of left ideals of R satisfying the following conditions:

1. (σ1) If I ∊ Σ, J is a left ideal of I, and J ⊇ I, then J ∊ Σ.

2. (σ2) If I ∊ Σ and r ∊ R, then Ir-1 = {s ∊ R | sr ∊ I} ∊ Σ.

3. (σ3) If I is a left ideal of R, J ∊ Σ, and It-1 ∊ Σ for each t ∊ J, then I ∊ Σ.

If U is an open set in Euclidean N-space EN which has finite Lebesgue measure |U| then a complete packing of U by open spheres is a collection C={Sn} of pairwise disjoint open spheres contained in U and such that Σ∞n=1|Sn| = |U|. Such packings exist by Vitali's theorem. An osculatory packing is one in which the spheres Sn are chosen recursively so that from a certain point on Sn+1 is the largest possible sphere contained in (Here S- will denote the closure of a set S). We give here a simple proof of the "well-known" fact that an osculatory packing is a complete packing. Our method of proof shows also that for osculatory packings, the Hausdorff dimension of the residual set is dominated by the exponent of convergence of the radii of the Sn.

A (simple, undirected) graph G is vertex transitive if for any two vertices of G there is an automorphism of G that maps one to the other. Similarly, G is edge transitive if for any two edges [a, b] and [c, d] of G there is an automorphism of G such that {c, d} = {f(a), f(b)}. A 1-path of G is an ordered pair (a, b) of (distinct) vertices a and b of G, such that a and b are joined by an edge. G is 1-transitive if for any two 1-paths (a, b) and (c, d) of G there is an automorphism f of G such that c = f(a) and d = f(b). A graph is regular of valency d if each of its vertices is incident with exactly d of its edges.

In this note we shall present a result about incidence functions on a locally finite partially ordered set, a result which is related to theorems of Lambek [2] and Subbarao [6]. Our terminology and notation will be that of Smith [4, 5] and Rota [7].

Let (L, ≤) be a partially ordered set which is locally finite in the sense that for all x, y ∊ L the interval [x, y] = {z | x ≤ z ≤ y} is finite. Denote by A(L, ≤) the set of functions f from L × L into some field, which is fixed once and for all, such that f(x, y) = 0 whenever x ≰ y.

Let (P, ≥) be a p.o. set. The basis graph of (P, ≥) is defined to be the directed graph whose vertex set is P and in which the ordered pair 〈a, b〉 is an edge if and only if b covers a in (P, ≥).

Let D be a directed graph. All graphs considered in this note are finite and are free of loops and multiple edges.

By an (incomplete) r × s latin rectangle is meant an r × s array such that (in some subset of the rs cells of the array) each of the cells is occupied by an integer from the set 1, 2, …, s and such that no integer from the set 1,2, …, s occurs in any row or column more than once. This definition requires that r≦s. If r=s we will replace the word rectangle by square. It is easy to see that for any n≧2 there is an incomplete n × 2n latin rectangle with 2n cells occupied which cannot be completed to a n × 2n latin rectangle. In this paper we prove the following theorem.

Theorem 1. An incomplete n × 2n latin rectangle with 2n—1 cells occupied can be completed to a n × 2n latin rectangle.

By an L-algebra we mean a power-associative nonassociative algebra (not necessarily finite-dimensional) over a field F in which every subalgebra generated by a single element is a left ideal. An H-algebra is a power-associative algebra in which every subalgebra is an ideal. The H-algebras were characterized by D. L. Outcalt in [2]. Let Sα be the semigroup with cardinality α such that if x, y ∊ Sα then xy = y. Consider the algebra over a field F with basis Sα. Such an algebra is an L-algebra that is not an H-algebra unless Sα contains only one element. In this paper we will prove that an algebra A over a field F with char. ≠ 2 is an L-algebra if and only if it is either an H-algebra or has a basis Sα where α is the dimension of A.

The following lemma, which appears as Lemma 4 in [5], was used to determine certain multicoherence properties of inverse limits of continua.

Lemma. Let X denote the inverse limit of an inverse system {Xλ, fλμ, Λ} of compact Hausdorff spaces Xλ. If Xλ has no more than k components (where k < ∞ is fixed) for each λ ∊ Λ, then X has no more than k components.

In this paper we give a set theoretic analogue of this lemma and an extension which was suggested to the author by Professor F. W. Lawvere. An application to inverse limits of finite groups is then given.

In this note we define two concepts which can be thought of as a generalization of noetherian concepts.

The main result is as follows (Corollary A): If R is a ring whose countably generated (left) ideals are (left) principal, then R is a (left) principal ideal ring.

This result if obtained, more generally, for any (left) R-module and any regular cardinal ℵα (Corollary 1); a cardinal ℵα is regular whenever W(ℵα) = {ordinals γ | card γ < ℵα} has no cofinal subset of cardinality less than ℵα.

Greenleaf states the following conjecture in [1, p. 69]. Let G be a (connected, separable) amenable locally compact group with left Haar measure, μ, and let U be a compact symmetric neighbourhood of the unit. Then the sets, {Um}, have the following property: given ɛ > 0 and compact K ⊂ G, ∃ m0 = m0(ɛ, K) such that

A graph G is a couple (X, R) where X is a set, R ⊂ X × X. If G is an undirected graph without loops (R a symmetric irreflexive relation), we can interpret G as a couple (X, R), where R is a set of two-element subsets of X, i.e. . This interpretation is generalized in the notion of society.

A society is a couple (X, R), where ; a k-society is a society (X, R) with |A| = k for each A ∊ R.

It is well known that if a module M is expressible as a direct sum of modules with local endomorphism rings, then such a decomposition is essentially unique. That is, if M = ⊕i∊IMi = ⊕j∊JNj then there is a bijection f: I → J such that Mi is isomorphic to Nf(i) for all i∊I (see [1]). On the other hand, a nonprincipal ideal in a Dedekind domain provides an example where such a theorem fails in the absence of the local hypothesis. Group algebras of certain groups over rings R of algebraic integers is another such example, where even the rank as R-modules of indecomposable summands of a module is not uniquely determined (see [2]). Both of these examples yield modules which are expressible as direct sums of two indecomposable modules in distinct ways. In this note we construct a family of rings which show that the number of summands in a representation of a module M as a direct sum of indecomposable modules is also not unique unless one has additional hypotheses.

In what follows, the transposed complex conjugate of a complex rectangular matrix D is denoted by D* and the rank of D by r(D). Meyer [1] proved the following result using generalized inverses:

Theorem. Let A and B be complex m × n matrices such that AB*=B*A=0. Then r(A+B) = r(A)+r(B).

Below we prove this result by repeated use of the fact that for every complex m × n matrix D we have r(D) = r(D*D) = r(DD*) (e.g. See [2] Theorem 5.5.4).

P. M. Cohn [7] calls a submodule P of the left A-module M pure iff 0 → E ⊗ P → E ⊗ M is exact for all right modules E. This concept has been studied in [11] and [12]. We will call a non-zero module pure simple iff its only pure submodules are 0 and itself, and the ring A left pure simple iff it is pure simple as a left A-module. We relate these concepts to the PP and PF rings of Hattori [13], and give several new characterizations of these rings. In order to establish these, we use the following known result: the Jacobson radical of any module is the sum of all its small submodules.

Parts of this paper are contained in the author's doctoral thesis [10] at McGill University.

A result proved by Favard for scalar-valued almost-periodic functions has an immediate extension to Banach space valued functions (see [2] and [3] for explicit details).

The result says that integrals of almost-periodic functions whose 'spectrum' is at positive distance from 0 are again almost-periodic. Our aim here is to indicate a more general formulation of this result using strongly almost-periodic oneparameter groups of operators in Banach spaces.