1·1. The main result. Let ø = ø(x1, x2, …, xn) be a word in n variables and let G be a group. A ø-element of G is any element of the form ø(x1, x2, …, xn)±1 with xi ∈G(1 ≤ i ≤ n). The subgroup generated by all the ø-elements of G is the verbal subgroup ø(G) of G. If there is a positive integer l such that every element of ø(G) can be expressed as the product of l or fewer ø-elements of G we say that G is ø-elliptic. If there is no such integer we say that G is ø-parabolic.

In an intersting article, ‘Bezout rings and their subrings’ (2) P. M. Cohn has introduced a construction which may also be applied in a slightly different situation, i.e. the imbedding of a Krull ring in a Dedekind ring.

Porte (1), p. 117, conjectures that the positive implicational propositional calculus has no finite characteristic matrix. The proof of this conjecture is a straightforward modification of Gödel's proof (2) that the intuitionistic propositional calculus has no finite characteristic matrix (see e.g. Church(3), ex. 26.12). Writing (A ∨ B) for ((A ⊃ B) ⊃ B) and Xij for (pj ⊃ pi) (i, j = 1,2,…), we define, for n > l, the formula

where the terms associate to the left. Since provable formulae take the value n for all systems of values of the variables in the matrix {1,…,n} where x ⊃ y is n when x ≤ y and y otherwise, whereas Gn takes the value n − 1 for the system of values pi = i (i = 1,…,n), it follows that Gn is not provable. On the other hand, since A ⊢ A ∨ B and B ⊢ A ∨ B, it is easily seen that is provable whenever r ≠ s (r, s = 1,…,n). The result follows from these two remarks.

In this paper I shall demonstrate certain algebraic properties of ideals of finite homological dimension in local rings.

In the first section, I show that no non-zero ideal of finite homological dimension in a local ring can be of zero grade (this is stated by Auslander and Buchsbaum in the appendix to (l), but I cannot find a proof in the literature). Using this result, together with the complex defined by a matrix, which is described by Eagon and Northcott in (2), I prove that an ideal of homological dimension one in a local ring Q may always, for some integer n, be described as the set of determinants of matrices obtained by adjoining to a certain (n–l)× n matrix with elements in the maximal ideal of Q another row with elements arbitrarily chosen in Q. (This result was established in (3), under the additional condition that Q should be a domain.)

Kruskal's theorem for obtaining a minimal (maximal) spanning tree of a graph is shown to be a special case of a more general theorem for matroid spaces in which each element of the matroid has an associated weight. Since any finite subset of a vector space can be regarded as a matroid space this theorem gives an easy method of selecting a linearly independent set of vectors of minimal (maximal) weight.

The importance of principal ideal domains (PIDs), both in algebra itself and elsewhere in mathematics is undisputed. By contrast, Bezout rings, ‡ although they represent a natural generalization of PIDs, play a much smaller role and are far less well known. It is true that many of the properties of PIDs are shared by Bezout rings, but the practical value of this observation is questioned by many on the grounds that most of the Bezout rings occuring naturally are in fact PIDs. However, there are several fairly natural methods of constructing Bezout rings from other rings, leading to wide classes of Bezout rings which are not PIDs, and it is the object of this paper to discuss some of these methods.

Suppose N is a set of points of a d-dimensional incidence space S and {Ha}, a ∈ I, a set of hyperplanes of S such that Hi ∈ {Ha} if and only if Hi ∩ N spans Hi. N is then said to determine {Ha}. We are interested here in the case in which N is a finite set of n points in S and I = {1, 2,…, n}; that is to say when a set of n points determines precisely n hyperplanes. Such a situation occurs in E3, for example, when N spans E3 and is a subset of two (skew) lines, or in E2 if N spans the space and n − 1 of the points are on a line. On the other hand, the n points of a finite projective space determine precisely n hyperplanes so that the structure of a set of n points determining n hyperplanes is not at once transparent.

In this note, I show that, if Q is a local ring and s an integer, 1 < s < codim Q, then there is an ideal As of Q, generated by three or fewer elements of Q, such that

and that, if Q is not regular, then there is an ideal B of Q, generated by three or fewer elements and of infinite homological dimension over Q.

The literature of finite projective planes consists largely of general investigations, taking in as many as possible of these systems at once. However, the geometry in a specific finite plane may well be an amusing, and not entirely trivial, field of study on its own. Some papers (5, 9, 13) have in fact appeared on the geometry of the translation plane of order 9: but the Hughes plane of the same order has comparatively been neglected. The object of the present paper is to make a beginning with the study of this plane from a synthetic point of view.

One of the most widely studied concepts in graph theory is that of the colouring of graphs. Much of the research in this area has dealt with the chromatic number χ(G) of a graph G, defined as the minimum number of colours which can be assigned to the points of G so that adjacent points are coloured differently. One can obtain a natural generalization of this concept as follows: we define the n-chromatic number χn (G) to be the smallest number of colours which one can associate with the points of G so that not all points on any path of length n are coloured the same. The 1-chromatic number is then simply the chromatic number.

In his discussion (3) of automorphisms of finite simple linear groups, Steinberg shows that the group denoted by B2(K) admits graph automorphisms when K is a perfect field of characteristic 2. The group B2(K) is identified in (2) as the symplectic group Sp4(K) when K is finite and of characteristic 2, and it is the object of this note to give, by means of this identification, an explicit description of the graph automorphisms of B2(2n).

Both Hirsch (cf. (5), (3)) and Eilenberg and Moore (cf. (7)) have described filtered complexes for the cohomology of a fibre space, under different hypotheses. Provided both complexes are defined, we shall give an explicit filtration-preserving map of the first into the second which induces homology isomorphisms. In section 3 we shall use this to transfer the action of the group in a principal bundle from the Hirsch complex (as computed by the author in (1)) to that of Eilenberg and Moore. The latter complex has certain advantages, for example, it is natural, and admits a cup-product structure.

Let Q be a quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined. If, for every finite sequence q1, q2, … of elements of Q, there exist i and j such that i < j and qi ≤ qj then we call Q well-quasi-ordered. For any ordinal number α the set of all ordinal numbers less than α is called an initial set. A function from an initial set into Q is called a transfinite sequence on Q. If ƒ: I1 → Q, g: I2 → Q are transfinite sequences on Q, the statement ƒ ≤ g means that there is a one-to-one order-preserving function ø:I1 → I2 such that f(α) ≤ g(ø(α)) for every α ∈ I1. Milner has conjectured in (3) that, if Q is well ordered, then any set of transfinite sequences on Q is well-quasi-ordered under the quasi-ordering just defined. In this paper, we define so-called ‘better-quasi-ordered sets’, which are well-quasi-ordered sets of a particularly ‘well-behaved’ kind, and we prove that any set of transfinite sequences on a better-quasi-ordered set is better-quasi-ordered. Milner's conjecture follows a fortiori, since every well ordered set is better-quasi-ordered and every better-quasi-ordered set is well-quasi-ordered.

In this paper we describe a group G such that for any simple coefficients A and for any i > 0, Hi(G; A) and Hi(G; A) are zero. Other groups with this property have been found by Baumslag and Gruenberg (1). The group G in this paper has cohomological dimension 2 (that is Hi(G; A) = 0 for any i > 2 and any G-module A). G is the fundamental group of an open aspherical 3-dimensional manifold L, and is not finitely generated. The only non-trivial part of this paper is to prove that the fundamental group of the 3-manifold L, which we shall construct, is not the identity group.

Let p be a prime and n ≥ 3. Then there is a simply connected CW-complex, , unique up to homotopy type, such that

Let X be a CW-complex. Write , the group of pointed homotopy classes of pointed maps from to X. The group structure derives from the fact that, under the restriction on n, is a suspension.

If Σn is a piecewise linear n-sphere and Qn is a piecewise linear n-ball which is a subpolyhedron of Σn thenis a piecewise linear n-ball.

Theorem. A simple closed curve J in the plane E2separates E2.

A topological ordered space (X, , <) is a set X endowed with both a topology and a partial order <, and is usually denoted by (X, ), it being understood that the symbol ≤ is used to denote all partial orders.

Since Thom first introduced the notion of the ‘dual’ of a Steenrod square, in (12), it has become apparent that calculation with such duals in the cohomology of, say, a simplicial complex X will often yield information about the impossibility of embedding X in Sn, for certain values of n. For example, the celebrated theorem that cannot be embedded in can easily be proved in this way. In this paper, we seek to generalize this method to any pair of extraordinary cohomology theories h* and k*, and natural stable cohomology operation θ: h* → k*. We show in section 3 that a simplicial embeddingf: X → Sn gives rise via the Alexander duality isomorphism to a homology operation

which is independent of n, the particular embedding f, and even the particular triangulations of X and Sn. If h* and k* are multiplicative cohomology theories, there are Kronecker products

if h0(S0) = k0(S0) = G, a field, and the Kronecker products make h*, h* and k*, k* into dual vector spaces over G, then can be turned into a cohomology operation c(θ): k*(X)→h*(X), by using this duality. This is certainly true if h* = k* = H*(;Zp), p prime, and in this case we have the original situation considered by Thom, who showed, for example, that

In order to study 3-manifolds with some particular group as fundamental group, one likes to have examples. There is an absence in the literature of means of constructing such examples. In this paper, theorems are proved which give necessary and sufficient conditions for the canonical 2-complex which corresponds to a group presentation to be a spine (2-dimensional skeleton in a cell decomposition with one 3-cell) of a connected closed† orientable 3-manifold. By enumerating all presentations of a group given by a particular presentation, two of the theorems provide an effective algorithm which permits one to attempt in a systematic way to construct 3-manifolds with a given group as fundamental group. Since every closed 3-manifold has a spine which corresponds to a group presentation, if the group is the fundamental group of an orientable 3-manifold, then the algorithm will eventually yield a 3-manifold, if not, then one is out of luck. There is no way out of this since Stallings has shown (1) that no algorithm exists which can, for every finitely presented group G, answer the question: Is G the fundamental group of a 3-manifold?