It is well-known that on an inverse semigroup S the relation ≦ defined by a ≦ b if and only if aa−1 = ab−1 is a partial order (called the natural partial order) on S and that this relation is closely related to the global structure of S (cf. (1, §7.1), (10)). Our purpose here is to study a partial order on regular semigroups that coincides with the relation defined above on inverse semigroups. It is found that this relation has properties very similar to the properties of the natural partial order on inverse semigroups. However, this relation is not, in general, compatible with the multiplication in the semigroup. We show that this is true if and only if the semigroup is pseudo-inverse (cf. (8)). We also show how this relation may be used to obtain a simple description of the finest primitive congruence and the finest completely simple congruence on a regular semigroup.

It is well known that the square matrix, of rank n−k + 1,

which we shall denote by B where any element to the left of, or below the nonzero diagonal b1, k, b2, k + 1, . …, bn−k + 1, n is zero, can be resolved into factors Z−1DZ; where D is a square matrix of order n having the elements d1, k, d2, k + 1, . …, dn−k + 1, n all unity and all the other elements zero, and where Z is a non-singular matrix. In this paper we shall show in a particular case that this is so, and in the case in question we shall exhibit the matrix Z explicitly. Application of this is made to find the classical canonical form of a rational integral function of a square matrix A.

Varieties of topological groups have been investigated in several papers ((2) and (10)-(13)). In this note we investigate the varieties generated by classical Lie groups. In particular we show results of which the following is indicative: The variety generated by the unitary group U(n) contains U(m) if and only if m ≦ n. En route we introduce the notion of a variety of topological Lie algebras which provides a convenient setting in which to answer our questions.

The number of groups of n which may be selected from 2n is 2n(2n − l)…(n + l)/n! But make the 2n into two groups of n, and select r out of the first and n − r out of the second. This gives [n(n−l)…(n−r+l)/r!] + [n(n−l)…(r + l)/(n−r)!] ways of thus making a group of n. Hence

The main result is a theorem giving several possibilities for the action of a 2-generator group acting on a Λ-tree, generalising the result that, if the action is free then the group is either free or free abelian. This involves investigation of several cases in which the action is shown to be properly discontinuous. This leads to a generalisation of results of Culler and Morgan, characterising abelian, dihedral and irreducible actions on ℝ-trees, to arbitrary Λ-trees.

The remark of Carnot quoted by Dr Mackay at the last meeting, regarding the mutual dependence of the four equations derivable from the same figure in the case of each of these theorems, led me to consider more fully the mutual dependence of the equations in question; and the following are some of the results arrived at.

In this paper we give a differential characterization of homogeneous Kähler submanifolds of complex projective spaces in terms of the existence of a tensor field, the homogeneous structure S. We show that for any m∈M, Sm determines a unitary representation whose orbit at m is a compact, complete Kähler submanifold which extends M. We consider the U(n) × U(N ~ n) (n = dim ℂM) module of the space of these tensors and we find its irreducible factors.

The following proof of the fundamental Combination Theorem does not appear in any of the current text-books on Algebra. It has the twofold advantage of being exceedingly simple and of being quite independent of the fundamental Permutation Theorem.

It is known that ℚ-derived univariate polynomials (polynomials defined over ℚ, with the property that they and all their derivatives have all their roots in ℚ) can be completely classified subject to two conjectures: that no quartic with four distinct roots is ℚ-derived, and that no quintic with a triple root and two other distinct roots is ℚ-derived. We prove the second of these conjectures.

AMS 2000 Mathematics subject classification: Primary 11G30

This note contains an elementary proof of the well-known result that every solution of Besse's equation

with a real parameter n has an infinite number of real positive zeros.

It is sometimes desirable to know in what circumstances a measurable set valued function admits a measurable selector; this problem occurs regularly in the theory of optimal control (see for example (3) and (7)). In this paper we demonstrate the existence of measurable selectors in two particular cases where the choice of selector has a simple geometrical interpretation, namely that of being a “ nearest-point ” selector, as is explained in detail below. This work derives in part from that of C. Castaing, particularly from Théorème 3.4 of (2), of which this is an extension.

Sufficient conditions for the absolute summability (A)1 of a Fourier series have been given by J. M. Whittaker and B. N. Prasad. They obtained theorem 1 below in the cases α = 0 and α = 1 respectively. Theorem 1 is contained in theorem 2, which was given by Prasad in the case α = 0.

Given a function f meromorphic in the plane, and complex numbers w and a, call w an a-point of f(k) if f(k)(w) = a. Denote by Λ(a,f) the set of z∈ℂ such that every neighborhood of z contains a-points of infinitely many of the functions f(k). Adapting the terminology of Pólya [16], who introduced the sets Λ(a,f) in [15],we call Λ(a,f) the final set of f with respect to a.