Analogues of the Krein-Milman theorem for order-convexity have been studied by several authors. Franklin [2] has proved a set-theoretic result, while Baker [1] has proved the theorem for posets with the Frink interval topology. We prove two Krein-Milman results on a large class of posets, with the open-interval topology, one for the original order and one for the associated preorder. This class of posets includes all pogroups. Cellular-internity defined in Rn by Miller [3] leads to another notion of convexity, cell-convexity. We generalize the definition of cell-convexity to abelian l-groups and prove a Krein-Milman theorem in terms of it for divisible abelian l-groups.

A well known theorem ([1] page 432) in the study of finite groups states that if P is a Sylow p-subgroup of the finite group G, and if P0 is a normal subgroup of P such that whenever two elements, σ and τ, of P are conjugate in G, the cosets σP0 and τP0 are conjugate in P/P0, then there is a normal subgroup K of G such that G = KP and K ∩ P = P0. In this note we will extend this result to allow P to be any Hall subgroup if G is solvable. More precisely, following theorem will be the proved.

Let S, T be self-mappings on a (non-empty) complete metric space (X, d). Let ai, i = 1, 2, …, 5, be non-negative real numbers such that < 1 and for any x, y in X,

A semiring (S, +,·) is a nonempty set S, endowed with associative operations of addition and multiplication, such that the multiplicative semigroup (S, ·) distributes over the addition. That is: x(y +z) = xy + xz and (x + y)z = xz + yz for all x, y and z in S. A topological semiring is a semiring, defined on a Hausdorff space, such that both of the operations are jointly continuous.

The aim of this note is to give an analogue, for an inverse semigroup S, of the theorem for a group G which says that if G is the set of normal subgroups of G, then the map N → (N) = {(a, b) ∈ G x G: ab-1 ∈ N}; for N ∈ G is a 1: 1 order preserving map of G onto ∧(G), the lattice of congruences on G. It will be shown that if E is the semilattice of idempotents of S, P = {E: α ∈ J} is a normal partition of E, and K is a certain collection of self conjugate inverse subsemigroups of S, then the map K →(X) = {(a, b)∈ S x S: a-la, b-1b∈ Eα for some α ∈ J and ab-l ∈ K) for K e Jf is a 1:1 map of K onto the set of congruences on S which induce P.

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.

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.

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.

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.

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.

In this paper we seek to establish some “complex inversion formulae” for the generalized Stieltjes transform for all s in the cut plane, supposing that p is any complex number except zero and the negative integers. The “cut plane” means all complex numbers except those which are negative real or zero.

The present paper investigates the thermal instability of a non-homogeneous fluid rotating between two co-axial cylinders when the inner cylinder is being heated uniformly. The conditions are established under which the oscillatory and non-oscillatory modes exist and further it has been shown that the oscillatory modes are amplified due to the adverse temperature gradient. In the case of non-oscillatory modes, sufficient conditions for stability and instability are obtained.

Let R be a ring with identity. We will use J(R) and P(R) to denote the Jacobson and prime radicals of R, respectively. If G is a group, the group ring of G over R will be denoted by RG.

We consider subspaces of a vector space UF, which is countably infinite dimensional over a recursively enumerable field F with recursive operations, where the operations in UF are also recursive, and where, of course, F and UF are sets of natural numbers. It is the object of this paper to investigate recursive equivalence types of such vector spaces and the ways in which their properties are analogous to and depend on properties of recursive equivalence types of sets.