In a previous paper [2], we gave an explicit description of the structure of all semirings with a completely simple additive semigroup. The next step is then clearly to consider semirings with a completely 0-simple additive semigroup. We are able to classify these semirings according to the multiplicative nature of their additive zero. Let R be a semiring whose additive semigroup is completely 0-simple with zero ∞. First, if ∞ ∞ ≠ ∞, then the multiplication of R is trivial. Besides these trivial semirings, another class of semirings with a completely 0-simple additive semigroup can be easily obtained by adjoining an element ∞ which is together an additive zero and a multiplicative zero.

Ever since David Ellis has shown that a Boolean algebra has a natural structure of an autometrized space, the interest in such spaces has led several authors to study various autometrized algebras like Brouwerian algebras [9], Newman algebras [4], Lattice ordered groups [6], Dually residuated lattice ordered semigroups [7] etc. However all these spaces are lattices (with the exception of Newman algebra which is not even a partially ordered set); and a natural question would be whether there are semilattices with a natural structure of an autometrized space. In the present paper we observe that the dual of an implicative semilattice [8] is a generalization of Brouwerian algebra and it has a natural structure of an autometrized space.

Let m, n and l be positive integers satisfying m ≦ n ≦ l ≦ 3. Denote by h(m, n, l) the largest integer with the property that from every n-subset of {1,2, …, m} one can select h(m, n, l) integers no l of which are in arithmetic progression. Let f(n, l) = h(n, n, l) and let g(n, l) = minmh(m, n, l). In what follows, by a P1-free set we shall mean a set of integers not containing an arithmetic progression of length l.

This paper, the second of a sequence beginning with [14], deals with the relationship between a distributive lattice L = (L; ∨, ∧, 0) with zero, and certain spaces of minimal prime ideals of L. Similar studies of minimal prime ideals in commutative semigroups [8] and in commutative rings [6] inspired this work, and many of our results are similar to ones in these two articles. However the nature of our situation enables many of these results to be pushed deeper and thus to arrive at a more satisfactory state; indeed with the insight obtained from the simpler lattice situation, one can return to some topics considered in [6], [8] and give complete accounts. We do not do this in the present paper, but leave the details to the reader, see e.g. [15]. Also a study of minimal prime ideals illuminates some topics in the theory of distributive lattices, particularly Stone lattices.

In [3] and [4], the near-rings R with no zero divisors are studied. In particular, a near-ring R is a near-field if it has a non-zero right distributive element ([4], Theorem 1.2.). Also, (R, +) is a nilpotent group if not all non-zero elements of R are left identities of R ([3], Theorem 2). The purpose of the present paper is to extend the above results to a class of near-rings with zero divisors; that is, the set of annihilators of an element x in R, T(x) = {g/xg = 0} is either {0} or R. The examples of such near-rings are those R with (R, +) simple groups and those R with no zero divisors as given in [1], [2], [3] and [4]. For this r, we can easily see that R = A∪S where A = {x/T(x) = R} and S = {x/T(x) = {0}}. Then the second part of this paper will give a structural theorem on the semi-group (S, · ), and more properties on R can be derived.

We introduce further finite groups which can be presented with an equal number of generators and relations.

Dickson's construction [1] of radical and semi-simple classes for certain abelian categories is a rather straightforward procedure in comparison with the methods traditionally used in more general situations. In §2 of the present paper we use a well-known characterization of the lower radical class to obtain, via consideration of maps with accessible images, a similar “homomorphic orthogonality” characterization of radical and semi-simple classes of associative rings. By substituting certain other subring properties for accessibility, we are then able to obtain simple constructions of various types of radical classes, including those which are strict in the sense first used by Kurosh [3] for groups.

In an article that appeared some years ago in this journal, Takács [1] gave a uniqueness criterion for the solution of the moment problem where the Br are given numbers and the Pk are sought, Pk ≧ 0, σkPk = 1. Takács showed that if Br < ∞ for all r and then the solution is unique. In addition, he gave an explicit formula for the solution in this case where q is any number satisfying q ≧ 0 and q > p2 — 1.

In [4] a concept of a weakly projectable vector lattice has been introduced. Stone vector lattices [3] and thus all special types of them, like Riesz [5], σ-complete and complete vector lattices are weakly projectable. Moreover C[0, 1] is weakly projectable but not Stone [4]. As we see the collection W of weakly projectable vector lattices is quite large. This explains to some extent the difficulty in producing examples of vector lattices which do not belong to W. In this note an example of a Banach lattice [1] which is not weakly projectable is described.

Let E be a topological vector space (over the real or complex field). A well-known geometric form of the Hahn-Banach theorem asserts that if U is an open convex subset of E and M is a subspace of E which does not meet U, then there exists a closed hyperplane H containing M and not meeting U. In this paper we prove, among other things, that if S is a left amenable semigroup (which is the case, for example, when S is abelian or when S is a solvable group, see [3, p.11]), then for any right linear action of S on E, if U is an invariant open convex subset of E containing an invariant element and M is an invariant subspace not meeting U, then there exists a closed invariant hyperplane H of E containing M and not meeting U. Furthermore, this geometric property characterizes the class of left amenable semigroups.

By an arc we mean a nondegenerate simply ordered set which is compact and connected in its order topology. A space X is rigid if the only homeomorphism of X onto itself is the identity map. Examples of rigid totally disconnected compact ordered spaces may be found in [1], [2], and [3]. It is the purpose of this note to prove the existence of an arc such that no two of its subarcs are homeomorphic. The proof makes use of arcs of large cardinality and the technique of inverse limit spaces.

Several classes of topological spaces which are either the infimum or supremum of a family of metrizable topologies in the lattice of topologies were given by Anderson [1]. In this paper we investigate the completion of the set of metrizable topologies in four distinct convergence structure lattices on an infinite set. We also observe the influence on these completions of the question regarding the existence of nonnormal ultrafilters which Čech [3] asserted could be neither provednor disproved using present mathematical axiom systems.

In general, a structure is called rigid if it admits only the trivial structure preserving transformations. Of course, what is trivial depends on the context. In [7] the authors understand by a rigid graph one which has the property that the only edge preserving transformation of the vertices is the identity map. In other contexts, however, it is convenient to regard the constant maps as trivial also. (See [2] where a topological space is given which admits as continuous transformations only the identity and the constants.) The purpose of this note is to construct graphs rigid in the later sense; we call them completely rigid.

In this paper, we consider factor covered graphs, which are defined basically as connected graphs in which every edge belongs to a 1-factor. The main theorem is that for any two edges e and e′ of a factor covered graph, there is a cycle C passing through e and e′ such that the edge set of C is the symmetric difference of two 1-factors.

It is well known that a wide range of Special Function Theory can be realized by considering unitary representations of certain topological groups.

In this approach it is very important to determine all irreducible continuous unitary representations of the group in question.

For the group of movements this problem was initiated by Vilenkin [6]. Rather restrictive conditions were imposed in this paper and while he returned to the problem in [7], it was still not solved in full generality (among other things the representation space was assumed separable). The first complete solution appears to have been given by Thoma [5]. Here, the method was to show each irreducible continuous unitary representation equivalent to a particular representation in a space of square integrable functions.

Stewartson [5] considered second class oscillations in a spherical shell in the presence of a toroidal magnetic field. He followed Hide [2] and supposed the toroidal field to be uniform.

The iterated equation of generalized axially symmetric potential theory [1] is the equation where, in its simplest form, the operator Lk is defined by the function f f(x, y) being assumed to belong to the class of C2n functions and the parameter l to take any real value. In appropriate circumstances, which will be indicated later, the operator can be generalized but as this can be done without altering the methods used, the operator will be taken in the form where r, θ are polar coordinates such that x = r cos θ, y = r sin μ = cosθ.

Investigations of closure operators on an involution poset T lead to a certain type of closure operators (so called c-closure operators) that are closely related to projections on T.

In [1], Owen gave sufficient conditions for the uniqueness of certain mixed problems having elliptic and hyperbolic nature for the ultrahyperbolic equation. Recently, Diaz and Young [2] has obtained necessary and sufficient conditions for the uniqueness of solutions of the Dirichlet and Neumann problems involving the more general ultrahyperbolic equation The purpose of this paper is to present corresponding uniqueness conditions for the Dirichlet and Neumann problems for the singular ultrahyperbolic equation for all values of the parameter, α −∞ > α > ∞. The symbol Δ denotes the Laplace operator in the variables x1, …, xm, Dj indicates partial differentiation with respect to the variable yj (1 ≧ J ≧ n), and the summation convention is adopted for repeated indices including (iu2), where denotes differentiation with respect to the variable xi.

In Neuwirth's book “Knot Groups” ([2]), the structure of the commutator subgroup of a knot is studied and characterized. Later Brown and Crowell refined Neuwith's result ([1], and we thus know that if G is the groups of a knot K, then [G, G] is either free of rank 2g, where g is the genus of K, or a nontrivial free product with amalgamation on a free group of rank 2g, and may be written in the form , where F is free of rank 2g, and the amalgamations are all proper and identical.