This paper is a sequel to one entitled “Varieties of topological groups”. The variety V of topological groups is said to be full if it contains every group which is algebraically isomorphic to a group in V For any Tychonoff space X, the free group F of V on X exists, is Hausdorff and disconnected, and has X as a closed subset. Any subgroup of F which is algebraically fully invariant is a closed subset of F. If X is a compact Hausdorff space, then F is normal. Let V be a full Schreier variety and X a Tychonoff space, then all finitely generated subgroups of F are free in V.

A β-variety V is one for which the free group of V on each compact Hausdorff space exists and is Hausdorff. For any β-variety V and Tychonoff space X, the free group of V exists, is Hausdorff and has X as a closed subset. A necessary and sufficient condition for V to be a β-variety is given.

The concept of a projective (topological) group of a variety V is introduced. The projective groups of V are shown to be precisely the summands of the free groups of V. A finitely generated Hausdorff projective group of a Schreier variety V is free in V.

In his book, Eine neue Methode in der Analysis und deren Andwendungen, P. Turán proved a number of new theorems given lower bounds for sums of powers. Since it was only his intention to demonstrate a new type of result, his bounds are by no means best possible nor are his proofs easily susceptible of improvement.

We generalise Turán's so-called Main Theorems to exponential sums with polynomial coefficients by a simple method involving only the evaluation and estimation of certain determinants. This approach gives in each case a result known to be asymptotically correct in the various exponents, and when specialised to the case of constant coefficients it provides in each case best-known results.

Our method moreover applies in more general circumstances and provided only that the determinants which arise can be conveniently estimated serves to provide lower bounds for other than exponential sums.

It is shown that, if U is a subvariety of the join of a nilpotent variety and a metabelian variety and if V is a variety with a finite basis for its laws, then UV also has a finite basis for its laws. The special cases U nilpotent and U metabelian have been established by Higman (1959) and Ivanjuta (1969) respectively. The proof here, which is independent of Ivanjuta's, depends on a rather general sufficient condition for a product variety to have a finite basis for its laws.

In this paper it is shown that an Hadamard design with each letter repeated once and only once can exist for 2, 4 and 8 letters only.

L.D. Baumert and Marshall Hall, Jr have found a design with four letters each repeated three times. Their design and the design on four letters each repeated once, found by J. Williamson, is the totality previously published.

In this paper the problem solved is that of unsteady flow of a viscous incompressible fluid between two parallel infinite disks, which are performing torsional oscillations about a common axis. The solution is restricted to high Reynolds numbers, and thus extends an earlier solution by Rosenblat for low Reynolds numbers.

The solution is obtained by the method of matched asymptotic expansions. In the main body of the fluid the flow is inviscid, but may be rotational, and in the boundary layers adjacent to the disks the non-linear convection terms are small. These two regions do not overlap, and it is found that in order to match the solutions a third region is required in which viscous diffusion is balanced by steady convection. The angular velocity is found to be non-zero only in the boundary layers adjacent to the disks.

We introduce the notion of a positively independent set of elements in an ordered module. With this concept we determine a necessary and sufficient condition which insures that on a strictly ordered module over a strictly ordered ring there exists a strict total order refining the given order. This generalizes a previous result of Fuchs, concerning the case of ordered abelian groups.

As an application, let R be a strictly ordered totally ordered ring and let M be the R-module of all mappings from a set I into R, with pointwise order; then this order on M may be refined to a strict total order.

In this note it is proved that for a certain class of infinite dimensional Banach algebras the set of singular elements (the non-units) is dense in the weak topology.

Let R be an associative ring with identity in which every element is either nilpotent or a unit. The following results are established. The set N of nilpotent elements in R is an ideal. If R/N is finite and if x ≡ y (mod N) implies x2 = y2 or both x and y commute with all elements of N, then R is commutative. Examples are given to show that R need not be commutative if “X2 = y2” is replaced by “xk = yK” for any integer k > 2. The case N = (0) yields Wedderburn's Theorem.

For the purpose of this paper a logic is defined to be a non-empty set of propositions which is partially ordered by a relation of logical implication, denoted by “≤”, and which, as a poset, is orthocomplemented by a unary operation of negation. The negation of the proposition x is denoted by NX and the least element in the logic is denoted by 0, we write NO = 1.

A binary operation “→” is introduced into a logic, the operation is interpreted as material implication so that “x → y” is a proposition of the logic and is read as “x materially implies y”. If material implication has the properties

11. (x → 0) = NX, 12. if x ≤ y then (z → x) ≤ (z → y), 13. if x ≤ y then x ≤ (y ≤ z)= x → z, 14. x ≤ {y → N(y → Nx)}, then the logic is an orthomodular lattice. The lattice operations of join and meet are given by x ∨ y = Nx → N(Nx → Ny) x ∧ y = N(X → N(x → y)) and, in terms of the lattice operations, the material implication is given by (x → y) = (y ∧ x) ∨ NX.

Moreover the logic is a Boolean algebra if, and only if, in addition to the properties above, material implication satifies 15. (x → y) = (Ny → Nx).

I.N. Herstein has shown that an associative ring in which the nilpotent elements are “well-behaved”, and such that every element satisfies a certain polynomial identity, is commutative. This result is generalized here. Specifically, it is shown that an alternative ring R which satisfies the following three properties is commutative:

(i) for x ∈ R, there exists an integer n(x) and a polynomial px (t) with integer coefficients such that xn+1p(x) = xn;

(ii) for a fixed positive integer m, a a nilpotent and b an arbitrary element of R, a - am commutes with b - bm;

(iii) for the same m, a and b, (ab+b)m = (ba+b)m and (ab)m = ambm.

Examples are given to show that all three properties are essential, and it is shown that for associative rings certain modified versions of these properties are individually enough to assure that the commutator ideal of the ring is nil.

In the present note, we offer a simple characterization of perfect rings in terms of their components and socle sequences, which is subsequently used to establish a one-to-one correspondence between perfect rings and certain finite additive categories. This correspondence is effected by means of a matrix representation, which describes the way in which perfect rings are built from local perfect rings.

An abstract orthogonality relation is defined, a closure operation and a corresponding lattice of closed sets are associated with it. Necessary and sufficient conditions are obtained for the orthomodularity of a sub-ortholattice of the lattice of closed sets.

The following form of the Hahn-Banach theorem is proved: Let X be a linear space over the complex semifield E and let f: S → E be a linear functional defined on a subspace S of X. If p: X → RΔ is a seminorm with the property that ∣f(s)∣ ≪ p(s) for all s in S, then f has a linear extension F to X with the property that ∣F(x)∣ ≪ p(x) for all x in X.