Let ƒ : X → Y be a map and let e′: Y → ΩΣY be the usual embedding. Then we prove the following results.

Theorem 1. cat ƒ = cat(e′ƒ), w cat ƒ = w cat(e′ƒ) if Y is an H-space.

Theorem 2. conil ƒ = w Σ cat(e′ƒ) ≦ Σ w cat(e′ƒ) ≦ w cat(e′ƒ), where Σ the suspension functor. If we take X = Y and ƒ= lx, this result yields conil X ≦ w cat e', a result due to Ganea, Hilton, and Peterson(4).

Theorem 3. Suppose that Y is (m– 1)-connected and

Then conil ƒ = w Σ cat (e′ƒ) = Σ w cat(e′ƒ) = w cat(e′ƒ).

We give an example which answers affirmatively the following problem posed by Hofmann and Mostert (1, p. 200):

P. 10. Let S be a compact semigroup with identity 1, zero 0, and totally ordered -class space . Suppose that there is a one-parameter semigroup containing 0 and 1. Can S have any regular -classes aside from D(0) and D(1)?

I am indebted to Professor K. H. Hofmann and Professor P. S. Mostert for their many helpful suggestions.

1. We first give a method of embedding a given semigroup S in a semigroup R so that in the commutative diagram

the bottom map is an isomorphism of partially ordered spaces and some non-regular -class of S is embedded into a regular -class of R.

Suppose throughout that α >0, β is real, and Nis a non-negative integer such that αN+ β> 0. A series of complex terms is said to be summable (B, α,β) to l if, as x→ ∞,

where sn= a0 + a1 + … + an.The Borel-type summability method (B, α, β) is regular, i.e., all convergent series are summable (B, α,β) to their natural sums; and (B,1, 1) is the standard Borel exponential method B.

Our aim in this paper is to prove the following Tauberian theorem.

THEOREM. Iƒ

(i) p ≧ – ½, an = o(np), and

(ii) is summable (B, α,β) to l, then the series is summable by the Cesaro method(C, 2p + 1) to l.

Although the Lebesgue dimension function is topologically invariant, the dimension-theoretic properties of a metric space can sometimes be made clearer by the introduction of a new, topologically equivalent metric. A considerable amount of effort has been devoted to the problem of constructing such metrics; one example of the fruits of this research is the following theorem by Nagata (2, Theorem 5).

In order that dim R ≦ n for a metrizable space R it is necessary and sufficient to be able to define a metric p(x, y) agreeing with the topology of R such that for every ∊ > 0 and for every point x oƒ R,

imply

A metric ρ which satisfies the condition of this theorem is called Nagata's metric (this term was introduced, to the best of the author's knowledge, by Nagami (1, Definition 9.3)).

1. In this paper, X will always denote a locally compact Hausdorff space, C0(X) the algebra of all complex-valued continuous functions vanishing at infinity on X and B(X) the algebra of all bounded continuous complex-valued functions defined on X. If X is compact, C0(X) is identical to B (X) and all the results of this paper are obvious. Therefore, we will assume at the outset that X is not compact. If A represents an algebra of functions, AR will denote the algebra of all real-valued functions in A.

In this paper I consider the relations between the lattice of topologies on a fixed space and the lattice of σ-algebras on that space. It is found that the intersection of these two lattices is the lattice of complete Boolean algebras, and that this lattice is anti-atomically generated. Some sufficient conditions for a topology to contain a maximal σ-algebra are noted.

The problem in which we are interested is the following. Call an additively written group G finitely decomposable if G = Σ Gi is the weak sum of finite groups Gi, Consider the following property.

Property P. Each subgroup of G having cardinality less than G is contained in a finitely decomposable direct summand of G.

Does Property P imply that G is finitely decomposable? We shall demonstrate that the answer is negative even in the commutative case. Our question is closely related to (1, Problem 5). In (4), an abelian group is called a Fuchs 5-group if every infinite subgroup of the group can be embedded in a direct summand of the same cardinality. The question of whether or not a Fuchs 5-group is in fact a direct sum of countable groups has been open for several years.

A directed graphG is a set of vertices V and a subset of V × V called the edges of G. A path in G of length k,

is such that (vi, vi+1) is an edge of G for 1 ≦ i ≦ k. A directed graph G is strongly connected if there is a path from every vertex of G to every other vertex. A circuit is a path whose two end vertices are equal. An elementary circuit has no other equal vertices. See (1) for a fuller discussion.

Let G be a finite, strongly connected, directed graph (fscdg). The kth power Gk of G is the directed graph with the same vertices as G and edges of the form (i, j) where G has a path of length k from i to j.

Some well-known theorems on the representation of an ideal in a commutative ring as an intersection of ideals of a specified type remain valid when attention is restricted to certain subclasses of the ideals of the ring. For example, a homogeneous ideal in a graded Noetherian ring is the intersection of homogeneous primary ideals, and Seidenberg (16) has recently proved a similar result for differential rings. Other examples are provided by theorems on the representation of certain differential and difference ideals as intersections of prime differential or difference ideals.

Kolchin (6) began the development of a general theory applicable to such phenomena. In this paper, I continue the development of that theory.

In §§ 1 and 2, I have followed Kolchin's definition and discussion of divisible systems with minor modifications and additions.

The modular function Mwas introduced by Perron in (6). M(ξ) (for irrational ξ) is denned by the property that the inequality

is satisfied by an infinity of relatively prime pairs (p, q)for positive d,but by at most a finite number of such pairs for negative d.We will write

for the continued fraction expansion of ξ ∈ (0, 1) and for any finite collection y1,…, ykof positive integers we will write

It is known (see 6) that

Where

A topological space is called “rigid” if its autohomeomorphism group is trivial. In (1), de Groot and McDowell showed that there are rigid, 0- dimensional spaces of arbitrarily high cardinality but left open the question of whether or not there are compact,rigid, 0-dimensional spaces of arbitrarily high cardinality, pointing out that an affirmative answer implies the existence of arbitrarily large Boolean rings with trivial automorphism groups. In this paper we construct a class of rigid, 0-dimensional spaces X αof arbitrary infinite cardinality and show that their Stone-Cech compactifications βX αare also rigid, thus answering the above question affirmatively.

I would like to thank S. W. Willard, J. R. Isbell, and the referee for their careful readings of preliminary versions of this paper.

Notation,p and q are generic symbols for prime numbers. N(H, p) denotes the number of primes q, not exceeding Hy which are primitive roots (mod p).

g(p) denotes the least positive primitive root (mod p).

g*(p) is the least prime primitive root (mod p).

v(m) denotes the number of distinct prime divisors of the integer m.

τk(m) is the number of ways of representing the integer m as the product of k integers, order being important.

π(x, k, r) is the number of primes p, not exceeding x, which satisfy p ≡ r(mod k); while π(x) denotes the total number of p ≦ x.

logm x denotes the mth iterated logarithmic function which is defined, when possible, by

In (5),I considered two-term conditions in π-exact couples, of which the exact couple of Federer (7) is an example. Let M(X, Y)be the space of all maps from X to Y with the compact-open topology. Our aim in this paper is to construct a π-exact couple , where Xis a finite-dimensional (in the sense of Lebesgue) metric space and , a certain (rather large) class of spaces. Specifically, is the class of all topological spaces Xwhich possess the following property (P).

(P) Let Y be a (possibly infinite) simplicial complex. There exists x0 ∈ X and y0 ∊ Y such that [X, x0]≃ [Y, y0].

In § 5 it will be seen that contains all CW complexes and all metric absolute neighbourhood retracts (ANR)s.

The group ring AG of a group G and a ring A is the ring of all formal sums Σg∈G agg with ag ∈ A and with only finitely many non-zero ag. Elements of A are assumed to commute with the elements of G. In (2), Connell characterized or completed the characterization of Artinian, completely reducible and (von Neumann) regular group rings ((2) also contains many other basic results). In (3, Appendix 3) Connell used a theorem of Passman (6) to characterize semi-prime group rings. Following in the spirit of these investigations, this paper deals with the complete ring of (right) quotients Q(AG) of the group ring AG. It is hoped that the methods used and the results given may be useful in characterizing group rings with maximum condition on right annihilators and complements, at least in the semi-prime case.

In this paper I continue the process, begun in (2), of reducing and weakening the axioms of congruence needed for absolute geometry. The congruence axioms Cl*–C4*, C4**, and C5a–C5c (frequently referred to below) can all be found in (2) and will not be quoted again here. (This paper should be read in conjunction with (2); any attempt to make it self-contained would result in the repetition of large parts of (2).) The notation of (2) will be used throughout the paper.

The main result here is that axiom C5c is unnecessary. This is shown in § 1. In § 2 I discuss three other points arising from (2).

Note added in proof.Since writing this paper, I have constructed examples of (a) Archimedean planes satisfying Cl*-C4* in which not all points are isometric, (b) non-Archimedean planes satisfying Cl*-C4*but not C4**,and (c) one-dimensional geometries in which 2.1 (with “plane” replaced by “line“) is false.

Stone lattices were (named and) first studied in 1957 (5). Since then, a great number of papers have been written on Stone lattices and a very satisfactory theory evolved. Despite the fact that all chains with 0, 1 as well as all Boolean algebras are Stone lattices, it turns out that many of the nice theorems on Boolean algebras have analogues, in fact, generalizations for Stone lattices. To give just two examples: the characterization of Boolean algebras in terms of prime ideals (Nachbin (6)) is generalized in (5) (see also (9)); Stone's representation theory (8) is generalized in (4); see also (2).

Using the triple associated with a Stone algebra L, as introduced in the first part of this paper (1), we will investigate certain problems concerning the structure of a Stone lattice.

The following topics will be discussed: prime ideals, topological representation, completeness, relative Stone lattices, and the reduced triple.

It is assumed that the reader is familiar with §§ 2–4 of (1). For the sake of convenience, we will write L= 〈C, D, ϕ〉 to indicate that 〈C, D, ϕ〉 is the triple associated with L, and whenever convenient we will write the elements of L as ordered pairs 〈x, a〉, as it is given in (1, § 4, the Construction Theorem).

Herstein and Small have shown (1) that nil rings which satisfy certain chain conditions are nilpotent. In particular, this is true for nil (left) Goldie rings. The result obtained here is a generalization of their result to the case of any nil subring of a Goldie ring.

Definition. Lis a left annihilator in the ring R if there exists a subset S ⊂ R with L = {x∈ R|xS= 0}. In this case we write L= l(S). A right annihilator K = r(S) is defined similarly.

Definition. A ring R satisfies the ascending chain condition on left annihilators if any ascending chain of left annihilators terminates at some point. We recall the well-known fact that this condition is inherited by subrings.

Definition. R is a Goldie ring if R has no infinite direct sum of left ideals and has the ascending chain condition on left annihilators.

In (8, § 3.2) Steinberg proved the following result.

THEOREM. Let K be a finite field, G′ a simple Chevalley group (“normal type1’) over K. Then every automorphism of G’ is the composite of inner, graph, field, and diagonal automorphisms.

For the meaning of these notions, see (8). Our aim in this note is to indicate how the Theorem may be extended to arbitrary infinite fields K, provided we replace G′ by the group denoted G in (5) and ĝ in (8). This amounts to proving the Theorem for automorphisms of G′ which are induced by automorphisms of G; when K is finite, Steinberg's results show that all automorphisms of G′ arise in this way. As Steinberg points out, the sole use made of the finiteness of K in his argument is in the proof of the following statement: Let U be the subgroup of G′ corresponding to the set of positive roots, and let σ be any automorphism of G′; then Uσ is conjugate to U in G′.

Let A be a uniformly closed subalgebra of C(X), the algebra of all complex-valued continuous functions on a compact Hausdorff space X. If A separates the points of X and contains the constant functions, A is called a function algebra. The algebra A is said to be strongly regular on X if it has the following property.

Property. For each f in A, each point x in X, and every , there is a neighbourhood U of x and a function g in A with g(y) = f(x) for all y in U and for all y in X.

That is, each function in A is uniformly approximate on X by functions in A which are constant near any point of X. Stated in terms of ideals, strong regularity means that, for each x, the ideal of functions vanishing in a neighbourhood of x is uniformly dense in the maximal ideal at x.