Let S and T be compact (topological) semigroups, A be a closed subsemigroup of S, and f be a continuous homomorphism of A onto T. It is a natural question to ask is there a compact semigroup Z containing T and a continuous homomorphism φ of S onto Z such that φ restricted to A is f?

In the category of compact Hausdorff spaces, the answer to the analogous question may be given by the following construction due to Borsuk. Let X and Y be compact Hausdorff spaces, A be a closed subspace of X and f a continuous function of A onto Y. Then R(f, X) = {(x, y)|f(x) = f(y) or x = y} is a closed equivalence relation on X. The adjunction space Z(f, x) is X/R(f, X). The purpose of this note is to investigate this construction in semigroups.

Two new simple groups have recently been discovered by Z. Janko. One of these groups has order 50,232,960. As a first step in showing that there is precisely one (up to isomorphism) simple group of order 50,232,960, the author proves in this paper the following result: If G is a non-abelian simple group of order 50,232,960, then the structure of the centralizer of an element of order two in G is uniquely determined.

In a note added on 21 April 1969 to this paper, the author announces that he has proved the uniqueness of the simple group of order 50,232,960.

In this paper, the concept of a fitted semigroup of binary relations is introduced. The binary operation for these semigroups is ordinary composition of relations. A particular type of homomorphism is investigated which maps a certain kind of subsemigroup of one fitted semigroup into another. The main result states that these homomorphisms are injective and gives a representation for them. This and several related results are then applied to the semigroup of all binary relations on a topological space which have a certain prescribed type of topological property. Compactness is one of these properties and the semigroup SK[X] of all compact binary relations on a Hausdorff k–space X is given special attention. It is shown that X and Y are homeomorphic if and only if a number of statements are true which relate the semigroups SK[x] and SK[y].For example, in order that X and y be homeomorphic it is both necessary and sufficient that some nonzero ideal of SK[X] be isomorphic to some nonzero ideal of SK[y].

Standard homological methods and a theorem of Harrison on cotorsion groups are used to prove the result mentioned.

In this note Z denotes an infinite cyclic group, Q the additive group of rational numbers, Zp ∞ a p–quasicyclie group, and Ip the group of p–adic integers.

Pascual Llorente proves in [3] that Ext(Q,z) is an uncountable group, and gives explicitly a countably infinite subset. Very little extra effort produces the result embodied in the title, as follows.

This paper is concerned with certain subsets of a finite extension K of the quotient field of an integral domain R. These subsets are contained in the integral closure of R in K and when R is integrally closed they are identical with it, but generally they need not even be rings. Various inclusion relations are studied and examples are given to show that these inclusions may be strict (with one exception which is still undecided).

Certain Hilbert spaces of functions with known smoothness are considered. The weights and mesh of the quadrature formulae which have least estimate of error with respect to the norm for functions in these spaces are found and their properties are discussed.

A result of Reilly and Scheiblich for inverse semigroups is proved true also for regular semigroups. For any regular semigroup S the relation θ is defined on the lattice, Λ(S), of congruences on S by: (ρ, τ) ∈ θ if ρ and τ induce the same partition of the idempotents of S. Then θ is a congruence on Λ(S), Λ(S)/θ is complete and the natural homomorphism of Λ(S) onto Λ(S)/θ is a complete lattice homomorphism.

Every ring is isomorphic to a subdirect sum of subdirectly irreducible rings. Unfortunately, however, as is shown, the property of being subdirectly irreducible is not preserved under homomorphisms. An example is given of a finite non-commutative subdirectly irreducible ring R with heart (= the intersection of all non-zero ideals) H, such that R/E is isomorphic with GF(2) + GF(2). (GF(2) is the two element Galois Field.) Some additional properties of the ring R are listed and contrasts are made with results for commutative subdirectly irreducible rings; for example, the zero divisors of R do not form an ideal.

In the usual Krull-Schmidt-Azumaya theorem in abelian categories it is essential that each of the direct summands has a local ring of endomorphisms. A partial answer is given here to the case where this last condition is not satisfied by the indecomposable direct summands. It is found that those summands with local endomorphism rings are determined up to isomorphism and cardinality of occurrence.

Let (X, S) be a measurable space and S be a σ-algebra of subsets of X. A nonempty class M is said to be a class of null sets if M ⊂ S, M is closed under countable unions of sets and E ∩ F ∈ M whenever E ∈ M and F ∈ S. It is possible to show that such concepts as absolute continuity, singularity and independence of measures can be studied simply by classes of null sets and that similar results can be obtained under the condition that each disjoint subclass of S – M is countable, denoted (S – M)C. If (X, S,μ) is a measure space then M = {E ∈ S: μ(E) = O} is a class of null sets of S and S – M the class of all sets of positive measure. We say that a measure μ has the property σ if there exists a sequence of totally finite measures on S such that σ(E) = for all E ∈ S. This property of measures is weaker than total σ–finiteness of measures. The main result of the present paper is as follows: Let (X, S,μ) be a measure space and M = {E ∈ S: μ(E) = O}. Then (S – M)C if and only if μ has the property σ.

An N-semigroup is a commutative, cancellative, archimedean semigroup with no idempotent element. This paper obtains a representation of finitely generated N-semigroups as the subdirect product of an abelian group and a subsemigroup of the additive positive integers.

The group ring of a finite abelian group G over the field of rational numbers Q and over the rational integers Z is studied. A new proof of the fact that the group ring QG is a direct sum of cyclotomic fields is given – without use of the Maschke and Wedderburn theorems; it is shown that the projections of QG onto these fields are determined by the inequivalent characters of G. It is proved that the group of units of ZG is a direct product of a finite group and a free abelian group F and the rank of F is determined. A formula for the orthogonal idempotents of QG is found.

Analogues are developed to the sum theorems in the dimension theory of metric spaces. It is shown that, within the class of metric spaces, any locally countable, σ-locally finite, or closure-preserving sum of finite-dimensional sets is countable-dimensional. Similar results are obtained under the more general hypothesis of countable-dimensional rather than finite-dimensional sets.

Before variational methods can be applied to the solution of an initial boundary value problem for a parabolic differential equation, it is first necessary to derive an appropriate variational formulation for the problem. The required solution is then the function which minimises this variational formulation, and can be constructed using variational methods. Formulations for K-p.d. operators have been given by Petryshyn. Here, we show that a wide class of initial boundary value problems for parabolic differential equations can be related to operators which are densely invertible, and hence, K-p.d.; and develop a method which can be used to prove dense invertibility for an even wider class. In this way, the result of Adler on the non-existence of a functional for which the Euler-Lagrange equation is the simple parabolic is circumvented.

Consider a stably stratified liquid, whose density varies exponentially with the vertical co-ordinate, that is bounded above by a free surface and below by a bed whose height depends on only one of the horizontal co-ordinates. Suppose that a gravity wave, that may be either a surface or an internal one, is travelling in a direction normal to the lines of constant depth. It is shown that if the frequency is below a certain value an infinite number of waves, all of the same frequency but having differing wave lengths, are generated and expressions for their amplitude are given in terms of the changes in depth which are assumed to be small.

Extending the results of An†onovskiĭ, Bol†janskiĭ, and Sarymsakov on semifield metric spaces, the authors define a regular semifield metric to be one in which the distance in the standard Tychonoff product representation of a point from a disjoint closed set is nonzero. It is shown that every completely regular topological space possesses a completely regular semifield metric and that there is an equivalent completely regular semifield metric for every semifield metric space. A normal semifield metric is defined to be one in which the distance between two disjoint closed sets is nonzero and it is shown that possessing a normal semifield metric is equivalent to being a normal topological space. Finally, Cauchy nets in semifield metric spaces are introduced leading to the notion of completeness. It is shown that a semifield metric space is complete iff every Cauchy net with the property that its directed set has cardinality less than or equal to the cardinality of the indexing set of the Tychonoff product representation of the semifield converges.

An inverse semigroup which is a union of groups is called Cliffordian. A semilattice E is called universally Cliffordian if every inverse semigroup having E as semilattice of idempotents is Cliffordian. It is shown that E is universally Cliffordian if and only if it is anti-uniform, that is, if and only if no two distinct principal ideals of E are isomorphic.

A semilattice E satisfying the minimum condition is anti-uniform if and only if it is a well-ordered chain. Examples are given of anti-uniform semilattices of more complicated types.

Let S be a Hausdorff topological semigroup and Cb,(S), Cc (S), the spaces of real valued continuous functions on S which are respectively bounded and have compact support. A regular measure m on S is r*-invarient if m(B) = for every Borel B ⊂ S and every x ∈ S, where tx: s → sx is the right translation by x. The following theorem is proved: Let S be locally compact metric with the tx's closed. Then the following statements are equivalent: (i) S admits a right invariant integral on Cc (S). (ii) S admits an r*–invariant measure, (iii) S has a unique minimal left ideal. The above equivalence is considered also for normal semigroups and analogous results are obtained for finitely additive r*–invariant measures. Also in the case when S is a complete separable metric semigroup with the tx's closed, the following statements are equivalent: (i) S admits a right invariant integral I on Cb(S) such that I(1) = 1 and satisfying Daniel's condition. (ii) S admits an r*–invariant probability measure. (iii) S has a right ideal which is a compact group and which is contained in a unique minimal left ideal. Finally, in order that a locally compact S admit a right invariant measure, it suffices that S contain a right ideal F which is a left group such that (B ∩ F)x = BX ∩ Fx for all Borel B ⊂ S.

Let H(C) be the group of homeomorphisms of the Cantor set, C, onto itself. Let p: C → M be a map of C onto a compact metric space M, and let G(p, M) be is a group.

The map p: C → M is standard, if for each (x, y) ∈ C × C such that p(x) = p(y), there is a sequence and a sequence such that xn → x and hn (xn) → y Standard maps and their associated groups characterize compact metric spaces in the sense that: Two such spaces, M and N, are homeomorphic if and only if, given p standard from C onto M, there is a standard q from C onto N for which G(p, M) = h−1G(q, N)h, for some h ∈ H(C) The present paper exhibits a structure theorem connecting these characterizing subgroups of H(C) and products of spaces: Let M1 and M2 be compact metric spaces. Then there are standard maps p: C → M1 × M2 and pi: C → Mi, i = 1, 2, such that G(p, M1 × M2) = G(p1, M1) ∩ G(p2, M2).

It is well-known that a boolean ring is commutative. In this note we show that a distributively generated boolean near-ring is multiplicatively commutative, and therefore a ring. This is accomplished by using subdirect sum representations of near-rings.