It is shown that every compact nonconnected semigroup (semiring) which has commuting congruences, has a nontrivial continous homomorphic image which is iseomorphic to a direct product of finite congruence free semigroups (semirings). (This extends parts of earlier work by Kaplansky (1947) on compact rings.) It is also shown that there is a possibly finer representation but onto a product of congruence free semigroups (semirings) known only to be compact Hausdorff. A number of the techniques used evolve from work of Professor Wallace, who retired in mid-1973, and to whom this paper is dedicated.

The purpose of this paper is an extension of a theorem of Hughes (1973). He showed: Let R be a ring which has a right artinian right quotient ring and let G be a group which has a (transfinite) ascending normal series with each factor either finite or cyclic, but only a finite number of finite factors. Then the group ring RG has a right artinian right quotient ring.

Let be an infinite r.e. repère, an infinite dimensional r.e. space such that ≦ L(). A condition is derived that is both necessary and sufficient for the existence of an infinite subset β ⊂ such that L(β)∪ is not an α-space. Examples which satisfy this condition are exhibited, proving that the class of α-spaces is not closed under intersections.

Holton (1973) introduced the following concept. A graph G is semi-stable if there exists a point v in G for which Γ(Gv) = Γ(G)v: where Γ(G) is the automorphism group of G, Gv is the graph G with v deleted and Γ(G)v is the subgroup of Γ(G) that fixes v. We say G is semi-stable at v. A partial stabilising sequence in Gs1, v2,…vk of its points such that Γ(G)v1v2…vi ═ Γ(Gv1v2…v1) for i = 1,2,…,k. If there exists a partial stabilising sequence in G for which k equals the number of points of G then G is said to be stable (Holton (1973a)). Most natation and terminology in what follows is explained in Harary (1969).

Burkill (1957) introduced extended Riemann-Stieltjes integrals of the form ∫baƒ(x)dkg(x)/dxk-1 to provide an alternative approach to the theory of distributions. We will denote Burkill's integral by (B)∫baƒ(x)dkg(x)/dxk-1 Burkill's paper (1957) partially motivated the study of the Riemann-Stieltjes integral in Russell (1970). He showed that the two integrals are not equivalent.

Let G be a finite group, K a field of characteristic p and Γ the group algebra of G over k. Let e = ∑g∈Gαgg, αg ∈k, be a primitive central idempotent of Γ; let supp e = {g ∈ G: αg≠0}. We provide a short proof of a slightly stronger version of Theorem 5 of Green (1968).

The property Pk(K = 0,1,…,n) is formulated. For k = 0, n this property reduces to conditions A and B defined by Kiguradze (1962) for a class of ordinary differential equations. Sufficient conditions are then given which guarentee that a class of delay differential equations of odd order possesses property Pk. The property PK is also seen to be useful in reducing the number of types of positve solutions of a related nonhomogeneous delay differential equation.

Numerical range theory for linear operators on normed linear spaces and for elements of normed algebras is now firmly established and the main results of this study are conveniently presented by Bonsall and Duncan in (1971) and (1973). An extension of the spatial numerical range for a class of operators on locally convex spaces was outlined by Moore in (1969) and (1969a), and an extension of the algebra numerical range for elements of locally m-convex algebras was presented by Giles and Koehler (1973). It is our aim in this paper to contribute further to Moore's work by extending the concept of spatial numerical range to a wider class of operators on locally convex spaces.

The technique for finding for asymptotic number of unlabled trees of various sorts was developed by pólya (1973) and perfected by Otter (1848). Modern presentations are available in the book of Harary and Palmer (1973; Chapter 9), and in the paper of Bender (to appear).

Let P be the algebra of polynomials in one inderminate x over the complex field C. Suppose ∥ · ∥ is a norm on P such that the coefficient functionals cj: ∑αix1 → αj (j = 0,1,2,…) are all continuous with respect to ∥·∥, and Let K ⊂ C be the set of characters on P which are ∥·∥-continuous. then K is compact, C\K is connected, and 0∈K. K. Let A be the completion of P with respect to ∥·∥. Then A is a singly generated Banach algebra, with space of characters (homeomorphic with) K. The functionals cj have unique extensions to bounded linear functionals on A, and the map a →∑Ci(a)xi (a ∈ A) is a homomorphism from A onto an algebra of formal power series with coefficients in C. We say that A is an algebra of power series if this homomorphism is one-to-one, that is if a ∈ A and a≠O imply cj(a)≠ 0 for some j.