An equivalence relation δ on a semigroup S is called a left congruence of S if (u, v) Ε λ implies that (su, sv) Ε λ for every s in S. With every set ℒ of pairwise disjoint left ideals (i. e. subsets L of S such that SL⊂EL), one can associate the left congruence {(u, v)|u = v or there exists an L in ℒ such that u Ε L and v Ε L}. Thus every nonempty left ideal is a left congruence class (i. e. an equivalence class of some left congruence). A left congruence has the form just described if and only if all its nontrivial classes (i. e. its classes containing at least two elements) are left ideals. Such a left congruence is called a Rees left congruence if there is at most one nontrivial class. The identity relation on S is a Rees left congruence since the empty set is a left ideal by definition.

In this note we consider transcendental entire functions

whose power series contain gaps, i.e.

where Λ = {λk} is a suitable set of positive integers. We denote the set of all such functions f(z) by E(Λ). As usual M(r) = M(r, f) denotes the maximummodulus of f(z) on the circle |z| = r. The order p and the lower order λ of f(z) are defined by

respectively.

Let X be a Banach space and K a convex subset of X. A mapping Tof K into K is called a nonexpansive mapping if | T(x) – T(y) | ≦ | x – y | for all x, yεK.

In a recent paper [5] one of us has given a sufficient condition to be satisfied by a given property of radical classes within a universal class w in order that, for any subclass ℳ of w, there should be a smallest radical class having the given property and containing ℳ. The sufficient condition is that the classof all radical classes with the given property can be characterised as the class of all radical classes fixed by an admissible function F (see Section 1 below). In this paper a necessary and sufficient condition is derived and the corresponding result for semi-simpleclasses is also presented. These results are given in Section 2.

For any fixed value of x, let denote the set of all positive integers with exactly k prime factors counted according to multiplicity, each prime factor being ≦ x. In an earlier paper [1] in this journal we posed the following problem. Let

Show the existence or non-existence of an integer K such that, if

then

We now show that such a K exists, and that in (2) there is strict inequality in each case.

Ito and Seidman in [5] define a BG space as a locally convex space in whichthere exists a bounded set with a dense span. In this note we extend the idea to a class of not necessarily locally convex linear topological spaces (l.t.s.). We note the link between the idea of a BG space and Weston’s characterization in [7] of separable Banach spaces. Finally we examine σ-BG spaces; here the bounded set in the definition of a BG space is replaced by the union of a sequence of bounded sets.

The numerical range of a bounded linear operator A on a complex Hilbertspace H is the set W(A) = {(Af, f): ║f║ = 1}. Because it is convex andits closure contains the spectrum of A, the numerical range is often a useful toolin operator theory. However, even when H is two-dimensional, the numerical range of an operator can be large relative to its spectrum, so that knowledge of W(A) generally permits only crude information about A. P. R. Halmos [2] has suggested a refinement of the notion of numerical range by introducing the k-numerical ranges

for k = 1, 2, 3, …. It is clear that W1(A) = W(A). C. A. Berger [2] has shown that Wk(A) is convex.

In this paper we give necessary and sufficient conditions on a commutative semigroup in order that it should have a maximal homomorphic image of one of the following types: (1) groups, (2) semigroups which are unions of groups and (3) pseudoinvertible semigroups, i. e. semigroups having the property that some power of each element lies in a subgroup of the semigroup.

Let Q(S) denote the maximal right quotient semigroup of the semigroup S as defined in [4]. In this paper, we initiate a study of Q(S) when S is a semilattice of groups. A structure theorem for such semigroups is given by Theorem 4.11 of [2].

Let R be a commutative ring with an identity element, E a (unitary) R-module, and x1, x2, …, xs elements of R. In these circumstances it is possible to form the Koszul complex† K(x1, x2, …, xs|E) of E with respect to x1, x2,…, xs and to investigate the implications, for E and xl, x2, …, xs, if certain of the homology modules of this complex vanish. This was first undertaken by M. Auslander and D. A. Buchsbaum [1]. Among the many results they obtain, the following [1, Proposition 2.8, p. 632] is of particular interest in connection with the present paper:

If R is Noetherian, E is finitely generated, and x1x2,…, xs belong to the Jacobson radical of R, then the statements

(a) x1, x2,…, xsis an R-sequence on E,

(b) HpK(x1, x2,…,xs⃒E) = O for all p > 0,

(c) H1K(x1, x2,…., xs,⃒E) = 0,

are all equivalent.

In this paper we are concerned with the representation theory of the symmetric groupsover a field K of characteristic p. Every field is a splitting field for the symmetric groups. Consequently, in order to study the modular representation theory of these groups, it is sufficient to work over the prime fields. However, we take K to be an arbitrary field of characteristic p, since the presentation of the results is not affected by this choice. Sn denotes the group of permutations of {x1, …, xn], where x1,…,xn are independent indeterminates over K. The group algebra of Sn with coefficients in K is denoted by Фn.

Consider the system of difference equations

in which the unknown x(t) is a complex m-vector, t is a real variable and a1, …, an are complex m × m matrices whose elements are functions of t, x(t), x(t+1), …, x(t+n – 1). A positive definite hermitian form V(x1x2, …, xn), with constant coefficients, is called a strong autonomous quadratic Lyapunov function (written strong AQLF) of (1) if there exists a constant K > 1 such that K2v(t+1) < v(t) for all non-zero solutions x(t)of (1), where v(t) = V(x(t), x(t+ 1), …, x(t+n —1)). The existence of a strong AQLF is a sufficient condition for the trivial solution x =0 of (1) to be globally asymptotically stable. It is a necessary condition only in the special case of an equation

The following result is well known in the theory of analytic functions; see [1].

Theorem A. Suppose that f(z) is an entire function of a complex variable z. Then f(z) satisfies the functional equation

where z = x + iy (x, y real), if and only if f(z) = aexp(sz), where a is an arbitrary complex constant and s is an arbitrary real constant.

In ([6]; pages 36–41), Lambek constructs the maximal ring of quotients Q(R) of a commutative ring R by denning a multiplication on Homr(D, R) where D ranges over all the dense ideals of R, and this generalizes the classical construction of ring of quotients, (cf. [6] for all the references on the subject.)

This programme is carried over, in the first section of this article, to the categoryof commutative reductive semigroups. Examples show that the maximal semigroup of quotients of a commutative monoid can be different from the classical one.

The present paper deals with relations between flat overrings and quotient rings. Weare mainly concerned with Richman's results [10] on flat overrings and withthose of Davis [2], Gilmer [3], Gilmer and Heinzer [4], Gilmer and Ohm [5], and Mott [8], on rings with the QR propertyand with the property (#) defined in Section 1. Some of their results are generalized, and it is shown that certain theorems, which at first glance seem to have nothing in common, are in fact particular cases of a single more general theorem.

Let (χ Σ, μ) be a probability space, so that X is a non-empty set, Σ is a Boolean a-algebra of subsets of X, and μ is a probability measure defined on Σ. If D Ε S is such that μ(D)≠0, then one traditionally associates with D a new probability measure μD, called the conditional probability measure determined by D, and defined by μD(E)= μ(D∩E)/μ(D), for all EΕΣ.

A semigroup S with 0 and 1 is termed completely right injective provided every right unitary S-system is injective. A necessary condition for a semigroup to be com-pletely right injective is given in [2]; namely, every right ideal is generated by an idempotent. An example in section 3 of this paper shows the existence of semigroups with 0 and 1 satisfying this condition which are not completely right injective. In [3], it is shown that the condition that every right and left ideal is generated by an idempotent is necessary and sufficient in the case that S is both completely right and left injective (called completely injective). Such a semigroup is an inverse semigroup with 0 whose idempotents are dually well-ordered.

The development of the theory of absolute integrals derives from certain key facts. Among them are:

(I) An integral is a positive linear functional on a vector lattice, which is continuous in a certain sense.

(II) A function equal almost everywhere to a summable function is itself summable.

(III) Every measurable function is the pointwise limit of a sequence of elementary step functions.

A device that often plays an important role in measure theory, but which has not beenfully exploited in the theory of abstract integrals is that of

(IV) the smallest class containing a given class and having a certain property

(such as being a σ-ring of sets). It is our purpose in this paper to examine the theory of abstract real-valued absolute integrals axiomatically, in such a way as to isolate and clarify the roles of (I) through (IV).

Using the definition of a Riemann surface, as given for example by Ahlfors and Sario, one can prove that all Riemann surfaces are orientable. However by modifying their definition one can obtain structures on non-orientable surfaces. In fact nonorientable Riemann surfaces have been considered by Klein and Teichmüller amongst others. The problem we consider here is to look for the largest possible groups of automorphisms of compact non-orientable Riemann surfaces and we find that this throws light on the corresponding problem for orientable Riemann surfaces, which was first considered by Hurwitz [1]. He showed that the order of a group of automorphisms of a compact orientable Riemann surface of genus g cannot be bigger than 84(g – 1). This bound he knew to be attained because Klein had exhibited a surface of genus 3 which admitted PSL (2, 7) as its automorphism group, and the order of PSL(2, 7) is 168 = 84(3–1). More recently Macbeath [5, 3] and Lehner and Newman [2] have found infinite families of compact orientable surfaces for which the Hurwitz bound is attained, and in this paper we shall exhibit some new families.