Any extension of a group A by a group B can be embedded in their wreath product A Wr B. Here we consider generalizations of this result for inverse semigroups.

Suppose S is an inverse semigroup and ρ0 is a congruence on S. We put T = S/ρ0 and denote the natural map from S to T by ρ. The kernel of ρ is the inverse image ETρ−1 of the semilattice ET of idempotents of T. First we show that if each ρ0-class of idempotents of S is inversely well-ordered, then S can be embedded in K Wr T, the standard wreath product of K and T. In general, not all elements of K Wr T have inverses. However, we can define a wreath product W(K, T) which is an inverse semigroup and which contains S when the previous condition holds. If ρ0 is idempotent-separating and S is 0-bisimple, K is the union of zero and a family of isomorphic groups. In this case, we can replace K by a single component group G of K, augmented by zero, and show that S can be embedded in W(G0, T). These results are analogous to the extension theories of D'Alarcao [1] and Munn [3] and they give conditions under which all inverse semigroup extensions of an inverse semigroup A by an inverse semigroup T are contained in a semigroup with structure depending only on A and T.

In this paper we consider differential equations of the form

where the coefficients Ai are holomorphic functions in a domain G1 × G2 ⊂ C × C. We restrict our attention to those equations for which it is possible to represent the solutions in the form

where g1(z1) and g2(z2) are arbitrary holomorphic functions in G1 and G2 respectively. The coefficients a1, k and a2, k depend on the given differential equation. Within the last ten years a number of publications have been devoted to this kind of representation of solutions.

Our main purpose is to describe those separated locally convex spaces which can serve as domain spaces for a closed graph theorem in which the range space is an arbitrary Banach space of (linear) dimension at most c, the cardinal number of the real line R. These are the δ-barrelled spaces which are considered in §4. Many of the standard elementary Banach spaces, including in particular all separable ones, have dimension at most c. Also it is known that an infinite dimensional Banach space has dimension at least c (see e.g. [8]). Thus if we classify Banach spaces by dimension we are dealing, in a natural sense, with the first class which contains infinite dimensional spaces.

Let R be a ring (with identity). We shall call R a local ring if R is aright noetherian ring such that the Jacobson radical M is a maximal ideal (and so is the only maximal ideal), and R/M is a simple artinian ring. A local ring R with maximal ideal M is called regular if there exists a chain

of ideals Mi of such that Mi–1/Mi is generated by a central regular element of R/Mi (1 ≦ i ≦ n). For such a ring R, Walker [6, Theorem 2. 7] proved that R is prime and n is the right global dimension of R, the Krull dimension of R, the homological dimension of theR-module R/M and the supremum of the lengths of chains of prime ideals of R. Such regular local rings will be called n-dimensional. The aim of this note is to give examples of regular local rings. These arise as localizations of universal enveloping algebras of nilpotent Lie algebras over fields and localizations of group algebras of certain finitely generated finite-by-nilpotent groups.

In this note we formally solve the following triple integral equations,

where f1(x), f2(x) and f3(x) are integrable for 0<x<α, α<x<β and β<x<∞, respectively, and the function g(λ) is assumed to satisfy sufficient conditions for the Fourier sine transform to exist. A special case of this system arose in a problem concerned with transistors.

Let ℱ be a smooth foliation of codimension p on a smooth manifold Mm. We can define ℱ by an atlas of coordinate charts (U, (x, y)), called leaf charts, where (x, y): U → Rm−p × Rp are coordinate functions for which the leaves of ℱ are given by y1 constant,…,yp constant, in U. Clearly, on the overlap of two such leaf charts (U, (x, y)) and (U′, (x′, y′)) we have a coordinate transformation of the form

If y′ is always affine in y, i.e.

where and Bi are constants, we shall say that ℱ is a transversally affine foliation. This notion is, in a sense, dual to that of affine foliation, see [2], in which x′ is affine in x and each leaf has an induced flat affine structure.

Let ℂn,n denote the space of n × n matrices with complex entries and let ℋn denote the set of n × n hermitian matrices. Given any matrix A∊ℂn,n, the Lyapunov transformation corresponding to A is defined by ℐA(H) = AH+HA*, where H∊ℋn. Let PSD(n) be the set of all n × n hermitian positive semidefinite matrices. Taussky [8, 9] raised the problems of determining

and

Both of these problems seem to be difficult.

The purpose of this paper is to exhibit various Q-algebras (quotients of uniform algebras) which are Jacobson radical. We begin by noting easy examples of nilpotent Q-algebras and Q-algebras with dense nil radical. Then we describe two ways of constructing semiprime, Jacobson radical Q-algebras. The first is by directly constructing a uniform algebra and an ideal. This produces a nasty Q-algebra as the quotient of a nice uniform algebra (in the sense that it is a maximal ideal of R(X) for some X ⊆ ℂ). The second way is by using results of Craw and Varopoulos to show that certain weighted sequence algebras are Q-algebras. In fact we show that a weighted sequence algebra is Q if the weights satisfy (i) w(n+1)/w(n)↓0 and (ii) (w(n+l)/w(n))∊lp for some p≧ 1, but may be non-Q if either (i) or (ii) fails. This second method produces nice Q-algebras which are quotients of rather horrid uniform algebras as constructed by Craw's Lemma.

This note establishes two statements from R. M. Fossum's review [1] of a paper by E. A. Magarian [2]. Firstly, if A → B is a pure homomorphism (of commutative rings) then A[[x1,…,xs]] → B[[x1,…, xs]] is pure. Secondly, if Rn → R is a directed family of pure homomorphisms then ∪ Rn → R is pure. A consequence is that if Rn → R is a directed family of pure homomorphisms and if R is Noetherian, then ∪ Rn[[x1,…,xs]]is Noetherian.

The object of this paper is to prove that the only non-trivial solution in positive integers of the equation of the title is X = 3, Y = 4.

Substituting x = 2X – 1, y = 2Y – 1 gives with a little manipulation

Recently, Levin and Saxon [5], De Wilde and Houet [2] defined the σ-barrelledness while Husain [3] defined the countable barrelledness and countable quasibarrelledness. It is well-known that barrelled spaces are countably barrelled, and countably barrelled spaces are σ-barrelled. It is natural to ask whether there is some condition for σ-barrelled (resp. countably barrelled) spaces to be countably barrelled (resp. barrelled). Using the concept of S-absorbent sequences of sets, we are able to give such conditions in Theorem 2.5 and Corollaries 2.6 and 2.7.

The α-regular classes of any finite group G are important since they are those classes on which the projective characters of G with factor set α take non-zero value, and thus a knowledge of the α-regular classes gives the number of irreducible projective representations of G with factor set α (see [4]). Here we look at the particular case of the generalized symmetric group Cm wr Sl. The analogous problem of constructing the irreducible projective representations of Cm wr Sl has been dealt with in [6] by generalizing Clifford's theory of inducing from normal subgroups, but unfortunately, it is not in general possible to determine the irreducible projective characters (and hence the α-regular classes) by this method.

Throughout this note we deal with a series Σan of positive terms. The following tests for the convergence of this series are well-known.

Test 1. (Ratio test). LetThen, if K < 1, Σanconverges, while if K > 1, Σandiverges.

Test 2. (Root test). LetThen, if K < 1, Σanconverges, while if K > 1, Σandiverges.

The definition of Ωf,a on p. 57 should read a

Let A1 and A2 be sup-norm algebras, each containing the constant functions. Let P(A1, A2) denote the set of bounded linear operators from A1 to A2 which carry 1 into 1 and have norm 1. Several authors have considered the problem of describing the extreme points of P(A1, A2). In the case where A1 is the algebra of continuous complex functions on some compact Hausdorff space, and A2 is the algebra of complex scalars, Arens and Kelley proved that the extreme operators in P(A1, A2) are exactly the multiplicative ones (see [1]). It was shown by Phelps in [6] that if A1 is self-adjoint, then every extreme point of P(A1, A1) is multiplicative. In [4], Lindenstrauss, Phelps, and Ryff exhibited non-multiplicative extreme points of P(A, A) and P(H∞, H∞), where A and H∞ are, respectively, the disk algebra, and the algebra of bounded analytic functions on the open unit disk D. The extreme multiplicative operators in P(A, A) were described in [6]. Rochberg proved in [8] that, if T is a member of P(A, A) which carries the identity on D into an extreme point of the unit ball of A, then T is multiplicative and is an extreme point of P(A, A). Rochberg's paper [9] is a study of certain extremal subsets of P(A, A), namely, those of the form K(F, G) = {T∊P(A, A):TF=G}, where F and G are inner functions in A. We proved in [5] that, if F is non-constant, then K(F, G) contains an extreme point of P(A, A).

In what follows, B(H) will denote the C*-algebra of all bounded linear operators on a Hilbert space H. Suppose we are given a C*-subalgebra A of B(H), which we shall suppose contains the identity operator 1. We are concerned with the existence of states f of B(H) which satisfy the following trace-like relation relative to A:

Our first result shows the existence of states f satisfying (*), when A is the C*-algebra C*(x) generated by a normaloid operator × and the identity. This allows us to give simple proofs of some well-known results in operator theory. Recall that an operator × is normaloid if its operator norm equals its spectral radius.

An inverse semigroup R is said to be reduced (or proper) if ℛ∩σ= i (where σ is the minimum group congruence on R). McAlister has shown ([3], [4]) that every reduced inverse semigroup is isomorphic with a “P-semigroup” P(G, , ), for some semilattice , poset containing as an ideal, and group G acting on by order-automorphisms; (and, conversely, every P-semigroup is reduced). In [4], he also found the morphisms between P-semigroups, in terms of morphisms between the respective groups, and between the respective posets.