In a recent paper [1] we showed that there is a (1,) -correspondence between the homomorphisms of an inverse semigroup S and its normal subsemigroups. The normal subsemigroup of S corresponding to and determining the homomorphism μ of S is the inverse image under μ of the set of idempotents of Sμ and is called the kernel of the homomorphism μ. The inverse image of each idempotent of Sμ is itself an inverse semigroup [1], and each such inverse semigroup is said to be a component of the normal subsemigroup determined by μ.

The closed graph theorem is one of the deeper results in the theory of Banach spaces and one of the richest in its applications to functional analysis. This note contains an extension of the theorem to certain classes of topological vector spaces. For the most part, we use the terminology and notation of N. Bourbaki [1], contracting “locally convex topological vector space over the real or complex field” to “convex space”; here we confine ourselves to convex spaces.

In what follows, groups are written additively, and commutators are denoted by brackets:

A group is metabelian if it satisfies the law

It is conceivable, though not plausible, that this law is equivalent to a law, or a set of laws, in only three variables, or even two. The present note shows that this is not the case.

In an earlier paper [1] on groups which are the products of two finite cyclic groups with trivial intersection, certain permutations, called “semi-special”, played a certain role. The permutation π of the numbers 1, 2,…, n is semi-special if† πn=n, and if, for every y ε [n],

is again a permutation, namely a power (depending on y) of π.

The recurrence formulae for the Bessel, Legendre, hypergeometric and other such functions can all be related to each other by means of the E-functions. In this paper it will be shown how, starting from known recurrence formulae for the hypergeometric function, others can be derived. The E-function formulae are deduced in § 2, and the others in § 3.

Following, for example, Kurošs [8], we define the (transfinite) upper central series of a group G to be the series

such that Zα + 1/Za is the centre of G/Zα, and if β is a limit ordinal, then If α is the least ordinal for which Zα =Zα+1=…, then we say that the upper central series has length α, and that Zα= His the hypercentre of G. As usual, we call G nilpotent if Zn= Gfor some finite n.

The main result of this series of papers is a theorem on the free product of groups (Theorem 1) which formed part of a doctoral thesis. This theorem has an immediate application to the word problem (Theorem 2). Usually the word problem refers to a finite system of generators and a finite number of defining relations, but in this context it is more natural to allow an infinite number of generators and defining relations. This (infinite) word problem is not solvable in general (Example 2).

In the equation

dis any positive integer which is not a perfect square. For convenience we shall consider only those solutions of (1) for which x and yare both positive. All the others can be obtained from these. In fact, it is well known that if (x0, y0) is the minimum positive integer solution of (1), then all integer solutions (x, y) are given by

and, in particular, all positive integer solutions are given by