Let S be a compact, topological semigroup with identity. Suppose D, L and R are the D, L and R classes of some x ∈ S. Let (L, α., L/H), (R, β, R/H), (D, γ, D/H) and (D, δ, D/R) by the fibre spaces gotten where α, β γ an δ are the natural maps. It is shown that (D, γ, D/H) has topologically the same structure as the fibre space associated with (L, α, L/H) by R. Also if (L, α, L/H) is locally trivial (locally a cartesian product) then so is (D, δ, D/R) and if both (L, α, L/H) and (R, β, R/H) are locally trivial then so is (D, γ, D/H).

Let R be a relation on a set X, and if A ⊂ X set RA = {x ∣ (x, α) ∈ R for some α ∈ A} and AR = {x ∣ (α, x) ∈ R for some α ∈ A}. Also A is called an antiset in case no two distinct elements of A are related. If A is a collection of antisets, then we generate a topology T(A) by taking sets of the form RA or AR (or X or ø) as subbasic open sets. Then conditions are given under which this topology satisfies separation axioms, or is compact or connected. For example. Theorem: Let A contain the singletons. If for each x ∈ X and y ∈ X \ x, there is a z ∈ X such that (x, z) ∈ R ((z, x) ∈. R) and (y, z) ∉ R ((z, y) ∉ R), then T(A) is a T1-topology. The conditions used to obtain compactness or connectedness are analogous to the conditions used to get the same properties for the order topology on a totally ordered set. Finally, by modifying the definition of T(A) slightly, we obtain conditions so that if X is a tree and R the cutpoint order, then T(A) is the original topology.

A new linear expression in σ(ν), ν = l, 2, …, n, which vanishes identically is established. A linear expression in σ(ν)'s has been found for σ3(n). A similar expression in σ3(ν)'s has been proved for σ7(n) also, Ramanujan's τ(n) = P24(n-1) is given in three different ways as linear expressions in σ2k+1(n) and σK(ν)'s with k = 1, 3, 5 respectively. Again, the coefficient p48(n-2) is expressed as a linear expression in σ11(v)'S and σ5(ν)'s. In establishing these results advantage is taken of the general theorem, also established, that the coefficients of the square of a power series whose coefficients satisfy a certain functional equation are expressible as linear functions of the latter coefficients.

It has been shown by D. Stephen that the number N of open sets in a non-discrete topology on a finite set with n elements is not greater than 3 × 2n-2.We show that for admissable topologies on a finite group N ≦ 2n/r, where r is the least order of its non-trivial normal subgroups. This is clearly a sharper bound.

It is proved that an orthocomplemented poset P is an orthomodular lattice if and only if it admits a suitable defined set of order preserving maps. These maps are called projections. They are, in fact, Just the projections of the Baer *-semigroup associated with the orthomodular lattice.

We establish several infinite classes of regular graphs with the property that any two distinct vertices have a fixed number of other vertices joined to both of them. The graphs are found by constructing their incidence matrices, which correspond to certain Hadamard matrices.

A weak logical structure is defined as a set of boolean propositional logics in which one can define common operations of negation and implication. The set union of the boolean components of a weak logical structure is a logic of propositions which is an orthocomplemented poset, where orthocomplementation is interpreted as negation and the partial order as implication. It is shown that if one can define on this logic an operation of logical conjunction which has certain plausible properties, then the logic has the structure of an orthomodular lattice. Conversely, if the logic is an orthomodular lattice then the conjunction operation may be defined on it.

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).

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.

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.

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.

A theorem of Gaschütz is used to prove: Let τ be a homomorphism of the distributively generated near-ring R with identity element and descending chain condition for left modules, τ have finite kernel, and U(R) be the group of units of R; then U(Rτ) = U(R)τ.

Furthermore it is shown that the finiteness condition for ker τ can be dropped in the case of R being a ring.

A ring R is π-regular (periodic) if for each element x of R there is n = n(x) SO that xn = xn.a.xn (xn = xn.1.xn) (a depending on x). Let R be an algebraic algebra over a commutative ring F With identity. In this paper we prove that if every π-regular image of the ring F is periodic, then R is periodic. This result applies in particular to the algebraic rings R (over the integers) considered by Drazin and to the algebraic algebras R over algebraically prime fields. It extends a result of Drazin on torsin-free algebraic rings and a generalization by this author of Drazin's result.

In the present note the equation yn = x1-m ym is reduced, under appropriate conditions, to a quadratic autonomous system of differential equations in the plane. In pursuance of this new approach, the main geometric features of this autonomous system are determined and a method of solving it is outlined.

The following sufficient condition is obtained for the uniform approximability of compact operators on a reflexive Banach space by operators of finite rank: if S is the unit ball of X and ø: X* → C(S) is the imbedding ø(x*)x = x*(x) then we require ø(X*) to be complemented in C(S).

We show that, whenever m, n are coprime, each subvariety of the abelian-by-nilpotent variety has a finite basis for its laws. We further Show that the just non-Cross subvarieties of are precisely those already known.

A variety of groups has the amalgam embedding property if every amalgam of two -groups can be embedded in a -group. In this note the author proves that if is a variety of exponent O which satisfies a law W(x1, X2,…, xt) but not W(x1, x2,… xt) then does not have the amalgam embedding property.

Let A be an algebra of formal power series in one indeterminate over the complex field, D a derivation on A. It is shown that if A has a Fréchet space topology under which it is a topological algebra, then D is necessarily continuous provided the coordinate projections satisfy a certain equicontinuity condition. This condition is always satisfied if A is a Banach algebra and the projections are continuous. A second result is given, with weaker hypothesis on the projections and correspondingly weaker conclusion.