Let be a variety of groups. We say that the variety (the group G) has a semigroup law if u(x1, …, xn) = ν(x1, …, xn) is a law in (in G), with u and ν different words in the free semigroup freely generated by x1,…,xn. It seems very difficult to determine under what conditions a variety has a semigroup law. All we can say in general is that if does have a semigroup law, then it is in fact characterized by its semigroup laws (Section III).

Our main result is that the fourth power of the determinant of an arbitrary m by m persymmetric matrix is identically equal to the determinant of a certain 2m by 2m skew-symmetric matrix. Having proved this, we derive several other identities involving powers of persymmetric determinants.

1. The concept of a unique factorization domain (UFD) has been defined, for commutative (e.g. (4) page 21) and non-commutative (1) integral domains. We take the theory a stage further here by defining a unique factorization ring (UFR), where throughout, a ring is understood to mean a commutative ring with identity, possibly containing proper zero-divisors.

Matroid theory was first studied by Whitney (1) as an abstract theory of linear independence in vector spaces. Recently its importance in graph theory has been noticed by Tutte (2), Edmonds (3) and Nash-Williams (4,5). Less interest has been shown in the extremely close relationship between matroids and incidence geometries. In this note we develop the more geometrical aspects of matroid theory, paying particular attention to the fundamental role of the hyperplanes of a matroid in this theory.

It is well known that every representation of the group Un of unitary matrices of order n × n is equivalent to a unitary representation (see e.g. Little-wood (6), ch. XI). Our object in the present paper is to discuss some properties of those representations, and to construct a specific unitary representation.

It is shown that a necessary and sufficient condition that two regular polynomial matrices T, U have relatively prime determinants is that the equation TX + YU = E, where E is a constant matrix, has a unique solution with the degrees of X, Y less than the degrees of U, T respectively. This is a generalization of a well-known theorem for scalar polynomials. An alternative form for the condition in terms of the non-vanishing of a determinant, corresponding to the resultant of two scalar polynomials, is also obtained together with an equivalent determinant of lower order.

Let the generalized hypergeometric function of one variable be denoted by

where (a)m is the Pochhammer symbol ((1, 3)).

A graph G is a pair (V{G), E(G)); V(G) is the set of vertices of G, and E(G) the set of edges of G; E(G) is a subset of the Cartesian product V(G) × V(G) (all pairs (υi, υj) where υi∈ V(G), υj ∈ V(G)). If υ is a vertex of G we shall write υ ∈ G. The graph-theoretic terminology used in this paper is that of Berge (1). We shall consider only graphs which are finite, undirected, and without loops.

We define an unbased H-space to be a pair (A, m) where A is a space and

is a map such that the maps La: x → m(a, x), Ra: x → m(x, a) are homotopy equivalences for all a ∈ A. This is the same as James's definition of an H'-space in (3); we follow his notation as far as possible. (A, m) is homotopy-associative if the maps m(m × 1) and m(l × m) are homotopic. A left (right) a-inverse (a ∈ A) is a map w: A → A such that the composition m(w × 1) d (w(1 × w) d) is homotopic to ka, the constant map to a. d denotes the diagonal map a → (a, a).

By a closed m-manifold we mean a C∞ compact connected m-dimensional manifold, without boundary. M will denote a closed m-manifold (m > 0), and En will denote Euclidean n-space. All embeddings will be C∞.

The subject of squared rectangles and squared squares was developed by Brooks, Smith, Stone and Tutte(1) using techniques based on the theory of electrical networks. In this note we shall treat the (apparently) more general topic of rectangulations using purely algebraic techniques. The starting point is the observation that the incidence matrices of the pair of dual networks associated with a squared rectangle in (l) can be more naturally derived as incidence matrices of the squaring (or rectangulation) itself. Further, the topological information contained in the networks is a simple consequence of Euler's formula applied directly to the rectangulation.

We shall point out a connexion between convolution (as defined in (5); see also (7)) and the Arens multiplication on the second dual of a Banach algebra (1, 2). This enables us to demonstrate in an easy way the existence of a commutative Banach algebra whose second dual is not commutative. Examples can also be found in (3, 9) and elsewhere.

Let X be a closed surface, i.e. a compact connected 2-manifold without boundary. If Gx denotes the group of all homeomorphisms of X to itself, and Nx is the normal subgroup consisting of homeomorphisms which are isotopic to the identity, then the quotient group Gx/Nx is called the homeotopy group of X and is denoted by ∧x.

The idea of constructing a space of functions taking values in a locally convex space E from a linear space of scalar valued functions is well known. We can, for example, define a space consisting of all E-valued functions φ(t) such that for all elements e′ of the dual E′ of E. Besides this construction there are others which arise in special cases. This idea has been used to obtain integrals of vector-valued functions (compare (2), Chapter III, § 4). Schwartz has also used it in his paper on differentiable vector-valued functions (9) whose main result is the famous kernel theorem, as well as in introducing vector-valued distributions. It is natural to expect that the space of vector-valued functions obtained will inherit some properties of the function space and the vector space E. Therefore one usually starts from some function space which has interesting properties.

In the previous paper(8), we considered a property of families of functions we termed. ‘B-equicontinuity’. It was shown that B-equicontinuity is stronger than the usual equicontinuity, and is weaker than the equicontinuity defined by Bartle (3). In this paper we consider the concept of B-equicontinuity on topological transformation groups. The net characterization of equicontinuity obtained in (8) is used in discussion. It is proved in (1) that if (X, T, π) is almost periodic, the transition group {πt|t ∈ T} is equicontinuous. One might wonder whether this conclusion can be strengthened to say that {πt|t ∈ T} is B-equicontinuous; we show here by an example that this is not true and a partial solution to this problem is given. Some relations between almost periodicity and B -equicontinuity are also discussed.

If (X, g) is a paranormed space, with paranorm g (see (2)), then we denote by X* the continuous dual of X, i.e. the set of all continuous linear functionals on X. If E is a set of complex sequences x = (xk) then E† will denote the generalized Köthe–Toeplitz dual of E

Let X be an F-space, not necessarily locally convex, with a quasi-norm ‖x‖ (see (7), p. 31). The series

of elements of X is commutatively convergent if and only if converges for every permutation {Yn}nεω of {Xn}nεω.

Let G be a locally compact topological group, and let μ be the left Haar measure on G, with μ the corresponding outer measure. If R' denotes the non-negative extended real numbers, B (G) the Borel subsets of G, and V = {μ(C):C ∈ B(G)}, then we can define ΦG: V × V → R' by

where AB denotes the product set of A and B in G. Then clearly

so that a knowledge of ΦG will give us some idea of how the outer measure of the product set AB compares with the measures of the sets A and B.

We propose an exceptionally short and effortless approach to the fundamental facts of the representation theory of compact groups on Hilbert spaces and expand our methods to prove some generalizations concerning representations on arbitrary locally convex vector spaces.

Goldman (4) conjectured that if Z is a linear set having the property that for every (Lebesgue) measurable real function f the set f−1[Z] is a measurable set, then Z must be a Borel set. I pointed out (2) that any analytic non-Borel set provides a counterexample, and Eggleston(3) showed that a set can have the property but be neither analytic nor even an analytic complement, for example, any Luzin set. As Eggleston mentions, in the construction of Luzin sets the continuum hypothesis is assumed (compare Sierpiński(6), Chapter II), and the question arises whether it can be dispensed with in his theorem. We shall show that a non-analytic set having Goldman's property can be constructed with the help of the axiom of choice alone, without the continuum hypothesis; the problem for analytic complements remains open. We shall also generalize one of Eggleston's intermediate results.