Bass [1] proved that if R is a left perfect ring, then R contains no infinite sets of orthogonal idempotents and every nonzero left R-module has a maximal submodule, and asked if this property characterizes left perfect rings ([1], Remark (ii), p. 470). The fact that this is true for commutative rings was proved by Hamsher [12], and that this is not true in general was demonstrated by examples of Cozzens [7] and Koifman [14]. Hamsher's result for commutative rings has been extended to some noncommutative rings. Call a ring left duo if every left ideal is two-sided; Chandran [5] proved that Bass’ conjecture is true for left duo rings. Call a ring R weakly left duo if for every r ε R, there exists a natural number n(r) (depending on r) such that the principal left ideal Rrn(r) is two-sided. Recently, Xue [21] proved that Bass’ conjecture is still true for weakly left duo rings.

If (L; ƒ) is an Ockham algebra with dual space (X; g), then it is known that the semigroup of Ockham endomorphisms on L is (anti-)isomorphic to the semigroup Λ(X; g) of continuous order-preserving mappings on X that commute with g. Here we consider the case where L is a finite boolean lattice and ƒ is a bijection. We begin by determining the size of Λ(X;g), and obtain necessary and sufficient conditions for this semigroup to be regular or orthodox. We also describe its structure when it is a group, or an inverse semigroup that is not a group. In the former case it is a cartesian product of cyclic groups and in the latter a cartesian product of cyclic groups each with a zero adjoined.

There are two well-known methods to build up algebras from given algebras, the direct and inverse limits, and a systematic account of these constructions may be found in [2]. It is known that every algebra can be represented as a direct limit of finitely generated algebras although in some cases the representation is trivial. Furthermore, Haimo [3] has established a certain inverse limit representation for the class of all infinite Boolean algebras which generalises, in actual fact, to the class of all infinite lattices with 1. The purpose of this note is to exhibit a certain nontrivial inverse limit representation which is peculiar to the class of infinite, complete Boolean algebras.

Let A = (aij) be an n × n complex matrix. The permanent of this matrix is

where the sum is taken over all permutations p of the set {1, …, n}.

In a recent paper [1] E. H. Lieb proved an interesting theorem (see below) which he applied to verify some conjectures of M. Marcus and M. Newman. The purpose of this note is to give a simple proof of Lieb's theorem.

The space of Schwartz distributions on the unit circle Г in the plane is topologically a considerable generalization of the space of regular, finite Borel measures on Г. However, the order structure of is usually taken to be the same as that of : there are no “positive” distributions which are not measures. This perhaps warrants consideration, since the order structure of generates its topology. In this paper we construct a system of order structures for which is a more natural complement in the intermediate stages to the topology of and which provides an interpretation of with its Schwartz topology as a quotient of a generalized base norm space V′. Where denotes the space of continuous functions on Г with its supremum norm topology, V′ is the dual of . The space contains the infinitely differentiable functions on Г with their usual topology, and (via the pointwise ordering on ) in its product ordering is realized as a generalized order unit space. Some consequences for harmonic functions are discussed.

For any group G, denote by φf(G respectively L(G)) the intersection of all maximal subgroups of finite index (respectively finite nonprime index) in G, with the usual provision that the subgroup concerned equals Gif no such maximals exist. The subgroup φf(G) was discussed in [1] in connection with a property v possessed by certain groups: a group G has v if and only if every nonnilpotent, normal subgroup of G has a finite, nonnilpotent G-image. It was shown there, for instance, that G/φf(G) has v for all groups G. The subgroup L(G), in the case where G is finite, was investigated at some length in [3], one of the main results being that L(G) is supersoluble. (A published proof of this result appears as Theorem 3 of [4]). The present paper is concerned with the role of L(G) in groups G having property v or a related property a, the definition of which is obtained by replacing “nonnilpotent” by “nonsupersoluble” in the definition of v. We also present a result (namely Theorem 4) which displays a close relationship between the subgroups L(G) and φf(G) in an arbitrary group G. Some of the results for finite groups in [3] are found to hold with rather weaker hypotheses and, in fact, remain true for groups with v or a. We recall that if a group has a it also has v ([2]Theorem 2) but not conversely. For example, G = (x, y: y-1xy = x2)has v but not a. It is a well-known result of Gaschütz ([8], 5.2.15) that, in a finite group G, if His a normal subgroup containing φ(G) such that H/φ(G) is nilpotent than His nilpotent. This remains true in the case where G is any group with v [1, Proposition 1]. Our first result is in a similar vein and is a generalization of Theorem 9 of [7] and Theorem 1.2.9 of [3], the latter of which states that, for a finite group G, if G/L(G) is supersoluble, then so is G.

R. A. Rankin [3] considered the problem of finding, for each integer n ≧ 3, a sequence of positive integers containing no n−term geometric progression. He constructed such sets Bn having asymptotic density

For example A3 ≑ 0·71975, A4 ≑ 0·8626, and An→1 as n → ∞.

The existence of r-regular graphs such that each edge lies in exactly t triangles, for given integers t < r, is studied. If t is sufficiently close to r then each such connected graph has to be the complete multipartite graph. Relations to graphs with isomorphic neighborhoods are also considered.

In this note homomorphisms of (2, 3, n) = 〈x, y: x2 = y3 = (xy)n = 1) into PSL3(q) are considered. Of particular interest is (2, 3, 7) whose finite factors are referred to as Hurwitz groups. It is known (see [3]) that for certain q, PSL2(q) is a Hurwitz group, so that one might suppose that PSL3(q) is a natural place to search for new Hurwitz groups. This intuition turns out to be ill-founded, for as we shall see all Hurwitz subgroups of PSL3(q) have already been discovered in [3].

Let k be an algebraically closed field. By an algebra is meant an associative finite dimensional k-algebra A with an identity. We are interested in studying the representation theory of Λ, that is, in describing the category mod Λ of finitely generated right Λ-modules. Thus we may, without loss of generality, assume that Λ is basic and connected. For our purpose, one strategy consists in using covering techniques to reduce the problem to the case where the algebra is simply connected, then in solving the problem in this latter case. This strategy was proved efficient for representation-finite algebras (that is, algebras having only finitely many isomorphism classes of indecomposable modules) and representation-finite simply connected algebras are by now well-understood: see, for instance [5], [7],[8]. While little is known about covering techniques in the representation-infinite case, it is clearly an interesting problem to describe the representation-infinite simply connected algebras. The objective of this paper is to give a criterion for the simple connectedness of a class of (mostly representationinfinite) algebras.

A longstanding open question in low dimensional topology was raised by J. H. C. Whitehead in 1941 [9]: “Is any subcomplex of an aspherical, two-dimensional complex itself aspherical?” The asphericity of classical knot complements [7] provides evidence that the answer to Whitehead's question might be “yes”. Indeed, each classical knot complement has the homotopy type of a two-complex which can be embedded in a finite contractible two-complex. This property is shared by a large class of four-manifolds; these are the ribbon disc complements, whose asphericity has been conjectured, and even claimed, but never proven. (See [4] for a discussion.) It is reasonable and convenient to formulate the following.

In Section 33 of [2], Bonsall and Duncan define an element t of a Banach algebra to act compactly on if the map a → tat is a compact operator on . In this paper, the arguments and technique of [1] are used to study this question for C*-algebras (see also [10]). We determine the elements b of a C*-algebra for which the maps a→ba, a→ab, a→ab + ba, a→bab are compact (respectively weakly compact), determine the C*-algebras which are compact in the sense of Definition 9, of [2, p. 177] and give a characterization of the *-automorphisms of which are weakly compact perturbations of the identity.

In this paper we study some questions proposed by B. Schein [8] regarding the semigroup of binary relations Bx for a finite set X: what is the ideal structure of Bx, what are the congruences on Bx, what are the endomorphisms of Bx? For |X| = nit is convenient to regard Bx as the semigroup Bn of n×n (0, l)-matrices under Boolean matrix multiplication.

The Dedekind η-function is defined by

where τ lies in the upper half plane ℋ = {tau;|Im(τ) > 0}, and x = e2πiτ. It is a modular form of weight ½ with a multiplier system. We define an η-product to be a function f (τ) of the form

where rδ ε ℤ. This is a modular form of weight with a multiplier system. The Fourier coefficients of η-products are related to many well-known number-theoretic functions, including partition functions and quadratic form representation numbers. They also arise from representations of the “monster” group [3] and the Mathieu group M24 [13]. The multiplicative structure of these Fourier coefficients has been extensively studied. Recent papers include [1], [4], [5] and [6]. Here we study the connections between the density of the non-zero Fourier coefficients of f(τ) and the representability of f(τ) as a linear combination of Hecke character forms (defined in Section 4 below). We first make the following definition.

In an earlier paper [5] of the author bisimple weakly inverse semigroups with partial identities were studied. The aim of this paper is to extend the results to a wider class of semigroups, viz: bisimple weakly inverse semigroups with partial right unitoids. It is found that an ℛ-class of weakly inverse semigroup is a right skew groupoid R = (R, P), where P is a right skew semigroup [5], P⊆R, and R is a partial semigroup satisfying certain conditions. When S is a bisimple weakly inverse semigroup with E the set of partial right unitoids, it can be shown that the ℛ-class R = (R, P) containing E, which is a right skew groupoid, satisfies the following:

(i) for any a, b ∈ R, there exists c ∈ R such that Pa ∩ Pb = Pc;

(ii) for any a ∈ R, there exists a left identity e of R such that (Pa ∩ P)e = Pa ∩ P.

Let ζ(s) = σn-s (Res >1) denote the Riemann zeta function; then, as is well known, , where Bm denotes the mth Bernoulli number, In this paper we investigate the possibility of similar evaluations of the Epstein zeta function ζq(s) at the rational integers s = k> 2. Let

be a positive definite quadratic form and

where the summation is over all pairs of integers except (0, 0). In attempting to evaluate ζq(k) we are guided by Kronecker's first limit formula [11]

where γ is Euler's constant,

is the Dedekind eta-function, and τ is the complex number in the upper half plane, ℋ, associated with Q by the formula

On the basis of (1.3) we would expect a formula involving functions of τ. This formula is stated in Theorem 1, (2.13).