If X and Y are Tychonoff spaces then the continuous function f mapping X onto Y is said to be compact (perfect, or proper) if it is closed and point inverses are compact. If h is a continuous function mapping X onto Y then by a compactification of h we mean a pair (X*, h*) where X* is Tychonoff and contains X as a dense subspace, and where h*: X*→Y is a compact extension of h. The idea of a mapping compactification first appeared in (7). In (1) it was shown that any compactification of X determines a compactification of h, and that any compactification of h can be determined in this way. This idea was then developed in (2) and (3).

The purpose of this note is to show the existence of a lattice ordered group G with a finite set of generators and a recursively enumerable set of defining relations such that there is no decision procedure to determine whether or not an arbitrary word in the generators reduces to the identity in G. In addition to the usual group-theoretic words, we may also use the two lattice operations ∨ and ∧ ; for example, a−l(b∨c) is a word in the generators a, b and c. At first sight it might appear that since we have an even greater harvest of words than in group theory and there exist finitely presented groups H (H has a finite number of generators and defining relations) with an insoluble word problem (no decision procedure to determine whether an arbitrary word in the generators reduces to the identity)—see (1), (2), (4) or (6)—the same would be true of lattice ordered groups. Unfortunately, such a naïve approach overlooks two salient points. First, the class of lattice ordered groups is strictly smaller than the class of all groups; second, there are certain relations connecting the lattice operations with the group operations which hold true for all lattice ordered groups. For example, a(b∧c) = ab∨ac and (a∨b)−1 = a−1∧b−1.

The purpose of this paper is to extend to locally convex spaces and to uncountable systems several well-known results concerning infinite series, biorthogonal sequences, and Schauder bases. Section 2 gives three extensions of the theorem of Orlicz (10) and Pettis (11) and some lemmas that will be needed later. The third section introduces the notions of a summability basis and a summability basis of subspaces, and two main theorems are proved, including a simplification of Retherford and McArthur's proof (12) of a theorem of Nikol'skiĭ (9). Section 4 investigates the positive cone of an uncountable biorthogonal system, particularly conditions equivalent to the regularity of this cone.

The three common definitions of local compactness require, respectively, each point to have a compact neighbourhood, a neighbourhood basis consisting of compact sets, or a closed compact neighbourhood. These definitions are equivalent in Hausdorff or in regular spaces but not in general (3, 7).

In a recent paper, E. T. Copson (2) proves the following result:

Theorem C.Letki >0 (i = 1, …, m), k1 +…+ km = 1, and the real sequence (an) satisfy the inequality

If (an) is bounded, then it must be convergent.

Much has been written on inequalities concerning positive definite matrices, but a new insight may be gained by examining inequalities from the standpoint of the inverse matrix. The standard inequality of Hölder can then be used in a more fruitful manner. This leads to some new results and a rediscovery of some known results.

It is common property in the theory of transformation semigroups that the presence of all the constant maps ensures that automorphisms are induced by a permutation of the underlying set. Essentially, this goes back to Malcev (2); it has been extensively generalised by Sullivan in (4). For semigroups which do not contain the constants (for example, all surjective transformations of a set, or all injections) there is, as yet, no similar result. The purpose of this note is to provide one.

The main aim of the present note is to compare C(X) and C(υX), the spaces of real-valued continuous functions on a completely regular space X and its real 1–1 compactification υX, with regard to weak compactness and weak countable compactness. In a sense to be made precise below, it is shown that C(X) and C(υX) have the same absolutely convex weakly countably compact sets. In certain circumstances countable compactness may be replaced by compactness, in which case one obtains a nice representation of the Mackey completion of the dual space of C(X) (Theorems 5, 6, 7).

A well-known fact is that every automorphism of the symmetric group on a set must be inner (whether the set is finite or infinite) unless the set has exactly six elements (4, § 13). A long-standing conjecture concerns the analogue of this fact for the group A(S) of all order-preserving permutations of a totally ordered set S. The group A(S) is lattice ordered (l-group) by defining, for f, g ∈ A (S), f ≦ g whenever xy ≦ xg for all x ∈ S. From the standpoint of l-groups, A(S) is of considerable interest because of the analogue of Cayley's theorem proved in (2), namely every l-group may be embedded in some A(S). Unlike the non-ordered symmetric groups, which must be highly transitive, A(S) is severely restricted if assumed transitive. Of special interest are those cases when A(S) is doubly transitive (relative to the order), since the building blocks (primitive) of all transitive A(S) are either doubly transitive or uniquely transitive. It is easily seen that each inner automorphism of A(S) must preserve the lattice ordering (that is, must be an l-automorphism). It is tempting to conjecture that every l-automorphism of A(S) must be inner. However, an easy counterexample is at hand. If S consists of two copies of the ordered set of integers, one entirely above the other, then A(S) is the direct product of two copies of the l-group of integers, which has an outer l-automorphism exchanging the two factors. The conjecture referred to above is that if A(S) is transitive on S, then every l-automorphism of A(S) is inner.

A real matrix is called non-negative (positive) if all its entries are non-negative (positive). Two matrices A and B are said to be cogredient if there exists a permutation matrix Q such that QAQT = B. A square non-negative matrix is called reducible if it is cogredient to a matrix of the form

where the blocks X and Y are square. Otherwise it is called irreducible.

In this note we prove some cyclic inequalities which are generalisations of known results. We shall assume throughout that ai+n = ai ≧ 0 for all i, that no denominator in the statement of a result vanishes and finally that p, m and q are positive integers. We shall also use A(i, m) to denote with the convention that A(i, 0) = 0. The most interesting of our results is probably Theorem 2 since, in the special case p = 1, m = 2, r = 0, it gives a lower bound of ⅓n for the Shapiro sum . Although it is by no means best possible, see (2), our method implicitly gives a really simple way of obtaining this lower bound which, incidentally, is an improvement on Rankin's original result (5).

Recently Scheiblich (7) and Munn (3), amongst others, have given explicit constructions for FIA, the free inverse semigroup on a non-empty set A. Further, Reilly (5) has investigated the free inverse subsemigroups of FIA. In this note we generalise two of Reilly's lesser results, and also characterise the surjective endomorphisms of FIA. The latter enables us to determine the group of automorphisms of FIA, and to show that if A is finite then FIA is Hopfian (a result proved independently by Munn (3)). Finally, we give an alternative proof of Reilly's main theorem, which uses Munn's theory of birooted trees.

Let be a category with finite products and a final object and let X be any group object in . The set of -morphisms, (X, X) is, in a natural way, a near-ring which we call the endomorphism near-ring of X in Such nearrings have previously been studied in the case where is the category of pointed sets and mappings, (6). Generally speaking, if Γ is an additive group and S is a semigroup of endomorphisms of Γ then a near-ring can be generated naturally by taking all zero preserving mappings of Γ into itself which commute with S (see 1). This type of near-ring is again an endomorphism near-ring, only the category is the category of S-acts and S-morphisms (see (4) for definition of S-act, etc.).

Let G be a finite group generated by n elements and defined by m relations, then G has a presentation, G = {x1, …, xn | R1, …, Rm} = F/R where F is free on generators x1, …, xn and R is the normal closure in F of R1, …, Rm. The deficiency of this presentation is n − m. Since G is finite the deficiency is non-positive and the deficiency of G is the maximal over the deficiencies of all presentations for G.

Let X be a topological space. Then we may define the fundamental groupoid πX and also the quotient groupoid (πX)/N for N any wide, totally disconnected, normal subgroupoid N of πX (1). The purpose of this note is to show that if X is locally path-connected and semi-locally 1-connected, then the topology of X determines a “lifted topology” on (πX)/N so that it becomes a topological groupoid over X. With this topology the subspace which is the fibre of the initial point map ∂′: (πX)/N→X over x in X, is the usual covering space of X determined by the normal subgroup N{x} of the fundamental group π(X, x).

The authors have recently treated (2) the problem of finding subsets E of the real line , of type Fσ, such that E–E contains an interval and the k-fold vector sum (k)E is of measure zero. Positive results can be obtained, for all k, on the basis of a recent theorem of J. A. Haight (3), following earlier partial results (1), (4) for k ≦ 7; and indeed in these cases the problem has a solution with E a perfect set. An analogous problem, apparently in most respects subtler than the first, is the following. Do there exist finite regular Borel measures μ on such that is absolutely continuous (where is the adjoint of μ) and the kth convolution power μk is singular? Both problems are of interest in the general context of elucidating the properties of the measure algebra or, more generally, M(G) for locally compact abelian G. The second problem may be regarded as an attempt to provide (at least one aspect of) a multiplicity theory for the first.

Let S denote a compact semitopological semigroup (i.e. the multiplication is separately continuous) and P(S) the set of probability measures on S. Then P(S) is a compact semitopological semigroup under convolution and the weak * topology (4). Let Γ be a subsemigroup of P(S) and where supp μ is the support of μ ∈P(S). In the case in which S is commutative it was shown by Glicksberg in (4) that S(Γ) is an algebraic group in S if Γ is an algebraic group. For a general semigroup S, Pym (7) considered Γ = {η}, η being an idempotent, and established that S(Γ) is a topologically simple subsemigroup of S, i.e. every ideal of S(Γ) is dense in S(Γ). In this note we prove that if Γ is a simple subsemigroup of P(S) (a semigroup is simple if it contains no proper ideal) which contains an idempotent then S(Γ) is a topologically simple subsemigroup of S. We also give an example to show that our conclusion (hence also Pym's) is best possible in the sense that S(Γ) is not simple in general

We are interested here in proving the existence of solutions to the (generalised) boundary value problem

where A is a continuous n×n matrix on R+ = [0, ∞), F is a continuous n vector on R+ × S (S = a suitable subset of Rn), T is a bounded linear operator defined on (or on a subspace of) C[R+, Rn], the space of all bounded and continuous Rn-valued functions on R+, and r is a fixed vector in Rn. There is an abundance of papers dealing with the problem ((I), (II)) on finite intervals, either in its full generality (cf., for example, (1), (2), (3), (4), (6)), or for special cases of the operator T. The reader is especially referred to the work of Shreve (7), (8) for such problems on infinite intervals for scalar equations. A series representation of the solutions is given by Kravchenko and Yablonskii (5). Most of our methods are extensions of the corresponding ones on finite intervals with some variations concerning the application of fixed-point theorems. Examples of interesting operators T are

where V(t), M, N are n×n matrices with V(t) integrable.

Throughout we consider operators on a reflexive Banach space X. We consider certain algebraic properties of F(X), K(X) and B(X) with the general aim of examining their dependence on the possession by X of the approximation property. B(X) (resp. K(X)) denotes the algebra of all bounded (resp. compact) operators on X and F(X) denotes the closure in B(X) of its finite rank operators. The two questions we consider are:

(1) Is K(X) equal to the set of all operators in B(X) whose right and left multiplication operators on F(X) (or on B(X)) are weakly compact?

(2) Is F(X) a dual algebra?

We shall start by recalling the definition and some basic properties of a convexity space; a topological convexity space (tcs) will then be a convexity space together with an admissible topology, and will be a generalisation of a topological vector space (tvs). After showing that the usual tvs results connecting the linear and topological properties extend to this new setting we then prove a form of the Krein-Milman theorem in a tcs.