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 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 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.

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).

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.

Let Fn be the free group on {ai: i ∈ Zn}, where the set of congruence classes mod n is used as an index set for the generators. Let φ be the permutation (1, 2, 3, …, n) of Zn and denote by θ the automorphism of Fn induced by φ, namely

The purpose of this paper is to impose conditions on a radical class P so that the P-radical of the ring of n × n-matrices over a ring A is equal to the ring of n×n-matrices over the ring P(A). In (1), Amitsur gave such conditions, but with the stipulation that the radical class P contained all zero-rings (rings in which all products are zero). In what follows, we shall be working within the class of associative rings.

Suppose X is a Banach space and J is the interval −∞<t<∞. For 1 ≦ p<∞, a function is said to be Stepanov-bounded or Sp-bounded on J if

(for the definitions of almost periodicity and Sp-almost periodicity, see Amerio-Prouse (1, pp. 3 and 77).

In a previous paper (9), we introduced the spaces Fp, μ of testing-functions and the corresponding spaces of generalised functions. For 1≦p≦∞, Fp(=Fp, 0) is the linear space of all complex-valued measurable functions φ defined on (0, ∞) which are infinitely differentiable on (0, ∞) and for which

for each k = 0, 1, 2, …. In symbols,

Bade, in (1), studied Boolean algebras of projections on Banach spaces and showed that a σ-complete Boolean algebra of projections on a Banach space enjoys properties formally similar to those of a Boolean algebra of projections on Hilbert space. (His exposition is reproduced in (7: XVII).) Edwards and Ionescu Tulcea showed that the weakly closed algebra generated by a σ-complete Boolean algebra of projections can be represented as a von Neumann algebra; and that the representation isomorphism can be chosen to be norm, weakly, and strongly bicontinuous on bounded sets (8): Bade's results were then seen to follow from their Hilbert space counterparts. I show here that it is natural to relax the condition of σ-completeness to weak relative compactness; indeed, a Boolean algebra of projections has σ-completion if and only if it is weakly relatively compact (Theorem 1). Then, following the derivation of the theorem of Edwards and Ionescu Tulcea from the Vidav characterisation of abstract C*-algebras (see (9)), I give a result (Theorem 2) which, with its corollary, includes (1: 2.7, 2.8, 2.9, 2.10, 3.2, 3.3, 4.5).

Let S be a compact Hausdorff space and let Φ: C(S)→E be a linear operator defined on the space of real-valued continuous functions on S and taking values in a (real) topological vector space E. Then Φ is called exhaustive (7) if given any sequence of functions fn ∈ C(S) such that fn ≧ 0 and

then Φ(fn)→0 If E is complete then it was shown in (7) that exhaustive maps are precisely those which possess regular integral extensions to the space of bounded Borel functions on S; this is equivalent to possessing a representation

where μ is a regular countably additive E-valued measure defined on the σ-algebra of Borel subsets of S.

Let Γ be an irreducible and non-singular curve in [n] (n ≧ 3) which is the complete intersection of n − 1 primals of order m (m ≧ 2) with a common “self-polar” simplex S: by this I mean that the rth polar of each vertex of S with respect to any one of the defining primals is the opposite face of S counted m−r times, for r = 1, 2, …, m − 1. The various such Γ constitute the curves of the title; they were encountered in (2). When m = 2, Γ is the intersection of n − 1 quadrics with a common self-polar simplex in the familiar classical sense.

The Fibonacci groups are a special case of the following class of groups first studied by G. A. Miller (7). Given a natural number n, let θ be the automorphism of the free group F = 〈x1, …, xn |〉 of rank n which permutes the subscripts of the generators in accordance with the cycle (1, 2, …, n). Given a word w in F, let R be the smallest normal subgroup of F which contains w and is closed under θ. Then define Gn(w) = F/R and write An(w) for the derived factor group of Gn(w). Putting, for r ≦ 2, k ≦ 1,

with subscripts reduced modulo n, we obtain the groups F(r, n, k) studied in (1) and (2), while the F(r, n, 1) are the ordinary Fibonacci groups F(r, n) of (3), (5) and (6). To conform with earlier notation, we write A(r, n, k) and A(r, n) for the derived factor groups of F(r, n, k), and F(r, n) respectively.