It is well known that the complete norm topology on a Banach algebra is not unique in general, though semisimplicity is sufficient (but not necessary) for uniqueness. In this note we consider a class of topological algebras of formal power series which have unique Fréchet space topology. The structure of these algebras in the Banach algebra case will be considered in a later paper.

We exhibit a closure operation which serves to define saturated formations of finite soluble groups.

In this note the notion of a free topological product Gα of a set {Gα} of topological groups is introduced. It is shown that it always exists, is unique and is algebraically isomorphic to the usual free product of the underlying groups. Further if each Gα is Hausdorff, then Gα is Hausdorff and each Gα is a closed subgroup. Also Gα is a free topological group (respectively, maximally almost periodic) if each Gα is. This notion is then combined with the theory of varieties of topological groups developed by the author. For a variety of topological groups, the -product of groups in is defined. It is shown that the -product, Gα of any set {Gα} of groups in exists, is unique and is algebraically isomorphic to the usual varietal product. It is noted that the -product of Hausdorff groups is not necessarily Hausdorff, but is if is abelian. Each Gα is a quotient group of Gα. It is proved that the -product of free topological groups of and projective topological groups of are of the same type. Finally it is shown that Gα is connected if and only if each Gα is connected.

It is proved that every subdirectly irreducible ring can be obtained as a proper homomorphic image of another subdirectly irreducible ring. An example of a subdirectly irreducible ring R with heart H is given for which i) R has a non-subdirectly irreducible homomorphic image, ii) R/H is an integral domain, and iii) the commutator ideal C(R) of R coincides with R.

We show that each abelian l–group G is a large l–subgroup of a minimal vector lattice V and if G is archimedean then V is unique, in fact, V is the l–subspace of (Gd)^ that is generated by G, where Gd is the divisible hull of G and (Gd)^ is the Dedekind-MacNeille completion of Gd. If G is non-archimedean then V need not be unique, even if G is totally ordered.

Suppose there exist a balanced incomplete block design with λ = 1 and an affine resolvable balanced incomplete block design, the two designs having the same replication number. Combining these designs we construct two strongly regular graphs. This is applied to give a new family of design graphs ((v, k, λ)-graphs). Finally, we show that for any prime power n there are two non-isomorphic design graphs with v = n2(n+2), k = n(n+1) and λ = n.

We consider integer matrices obeying certain generalizations of the incidence equations for (v, k, λ)-configurations and show that given certain other constraints, a constant multiple of the incidence matrix of a (v, k, λ)-configuration may be identified as the solution of the equation.

It is shown that a von Neumann regular ring R is left seif-injective if and only if every finitely generated torsion-free left R-module is projective. It is further shown that a countable self-injective strongly regular ring is Artin semi-simple.

Let H(C) be the group of homeomorphisms of the cantor set, C onto itself. Let p: C → M be a (continuous) map of C onto a compact metric space M, and let G(p, M) be {h ∈ H(C) | ∀x ∈ C, p(x) = ph(x)}. G(p, M) 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). That is, two compact metric spaces are homeomorphic if and only if they determine, via standard maps, the same classes of conjugate subgroups of H(C).

The present note exhibits two natural structure theorems relating algebraic and topological properties: First, if M = H ∪ K (H, K ≠ π) , compact metric, and p : C → M are given, then G(p, M) is isomorphic to a subdirect product of G(p, M)/S(p, H\K) and G(p, M)/S(p, K\H) where, generally, S(p, N) is the normal subgroup of homeomorphisms supported on p−1M . Second, given M and N compact metric and p : M → N continuous and onto, let M ≠ M − CID*α ≠ 0 , where {Dα}α ∈ A is the collection of non-degenerate preimages of points in N Then there is a standard p : C → M such that fp : C → N is standard and there is a homomorphism

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The equation of motion of charged incoherent matter, and hence of a test charge, in general relativity can be written as the geodesic equation of an affine connection. Here, the connection is chosen to satisfy a condition which has the form of one of the basic geometrical principles of Einstein's unified field theory. The symmetric part of the fundamental tensor of the geometry is chosen to be the metric tensor of general relativity; equations to be satisfied by the skew part, involving the electromagnetic field, are obtained. An alternative condition on the affinity is also considered.

We show

(i) that the Rayleigh-Ritz method is a practical procedure for obtaining approximations to the velocity potential for compressible flows, and

(ii) how to calculate an estimate of the error in such approximations.

We construct a family of groups with precisely NO full orders.

The purpose of this paper is to show that there exists an infinite family of non-invertible prime 2-component links in S3. It is then noted that the existence of such a family implies the existence of a family of non-invertible wild arcs in S3, which are tame modulo an endpoint.

An n × n–matrix on n signed variables is called Hadamard of Williamson type if each variable occurs exactly once in each row, and the inner product of any pair of distinct rows is zero. We show here that these matrices correspond in a natural way to rational formulas for products of sums of n squares, shown by Hurwitz to exist only for n = 1, 2, 4, and 8. Hurwitz' arguments contain an implicit proof that this correspondence is one-to-one (we show this directly) and hence that Hadamard matrices of Williamson type exist for orders 1, 2, 4 and 8 only.

Let p be a prime and let Q be a centre-by-finite p′-group. It is shown that the ZQ-modules which satisfy the minimal condition on submodules and have p–groups as their underlying additive groups can be classified in terms of the irreducible ZpQ-modules. If such a ZQ-module V is indecomposable it is either the ZpQ-injective hull W of an irreducible ZpQ-module (viewed as a ZQ-module) or is the submodule W[pn] of such a W consisting of the elements ω ∈ W which satisfy pnw = 0.

This classification is used to classify certain abelian-by-nilpotent groups which satisfy Min-n, the minimal condition on normal subgroups. Among the groups to which our classification applies are all quasi-radicable metabelian groups with Min-n, and all metabelian groups which satisfy Min-n and have abelian Sylow p-subgroups for all p.

It is also shown that if Q is any countable locally finite p'-group and V is a ZQ-module whose additive group is a p-group, then V can be embedded in a ZQ-module whose additive group is a minimal divisible group containing that of V. Some applications of this result are given.

Let S be a compact connected semigroup. If the decomposition space of S, under the action of a compact connected subgroup at the identity, is a plane continuum then this decomposition is a congruence.