The following result is well known (as usual, [x]denotes the integral part of x):

(A) Let α and β be positive irrational numbers satisfying

1

Then the sets [nα], [nβ], n= 1, 2, …, are complementary with respect to the set of all positive integers]see, e.g. (1; 2; 4; 5; 6; 7; 8; 10; 13; 14; 15; 16). In some of these references the result, or a special case thereof, is mentioned in connection with Wythoff's game, with or without proof. It appears that Beatty (4) was the originator of the problem.

The theorem has a converse, and the following holds:

(B) Let α and β be positive. The sets [nα] and [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers if and only if α and β are irrational, and (1) holds.

In this paper all representations are over the complex field K. The generalized symmetric group S(n, m) of order n!mn is isomorphic to the semi-direct product of the group of n × n diagonal matrices whose rath powers are the unit matrix by the group of all n × n permutation matrices over K. As a permutation group, S(n, m) consists of all permutations of the mn symbols {1, 2, …, mn} which commute with

Obviously, S (1, m) is a cyclic group of order m, while S(n, 1) is the symmetric group of order n!. If ci = (i, n+ i, …, (m – 1)n+ i) and

then {c1, c2, …, cn} generate a normal subgroup Q(n) of order mn and {s1, s2, …, sn…1} generate a subgroup S(n) isomorphic to S(n, 1).

In (2), M. Auslander and O. Goldman laid the foundations for the study of central separable algebras. For unexplained terminology and notation, see (2). Here we are interested in projective modules and the ideal structure of a central separable algebra A over some special commutative rings K. When K is a field, one consequence of Wedderburn's Theorem is that there is a unique (up to isomorphism) irreducible A-module. We show here that if K is a commutative ring with a finite number of maximal ideals (semi-local) and with no idempotents other than 0 and 1 or if K is the ring of polynomials in one variable over a perfect field, then there is a unique (up to isomorphism) indecomposable finitely generated projective A-module. An example in (3) shows that this result fails if one only assumes that K is a principal ideal domain.

This paper developed from, and complements, the paper by F. R. DeMeyer (see 6).

In the first section of this paper we note a correspondence between projective modules of a central separable R-algebra A and the two-sided ideals of central separable algebras in the same class as A in the Brauer group of R. When R has the property that rank one projective A-modules are free, this correspondence yields a bijection between isomorphism types of indecomposable projective A-modules and the isomorphism types of algebras in the Brauer class of A which are the analogue of division algebra components in the field case. This bijection was remarked on without proof by DeMeyer in (6).

Pursuing the ideas behind this correspondence, we consider the situation for a separable order A in a central simple algebra A over an algebraic number field, and obtain, by means of results involving the reduced norm, a generalization of DeMeyer's remark except when the division algebra component of A is a totally definite quaternion algebra (Theorem 3.3).

In 1955, J. Surányi and P. Turán (8) initiated the problem of existence and uniqueness of interpolatory polynomials of degrees less than or equal to 2n — 1 when their values and second derivatives are prescribed on n given nodes. This kind of interpolation was termed (0, 2)-interpolation. Later, Balázs and Turán (1) gave the explicit representation of the interpolatory polynomials for the case when the n given nodes (n even) are taken to be the zeros of πn(x) = (1 — x2)Pn′(x), where Pn–i(x) is the Legendre polynomial of degree n — 1. In this case the explicit representation of interpolatory polynomials turns out to be simple and elegant.

Balázs and Turán (2) proved the convergence of these polynomials when f(x) has a continuous first derivative satisfying certain conditions of modulus of continuity. They noted (1) that a significant application of lacunary interpolation could possibly be given in the theory of a differential equation of the form y′ + A(x)y= 0.

A well-known result of Dembowski and Wagner (4) characterizes the designs of points and hyperplanes of finite projective spaces among all symmetric designs. By passing to a dual situation and approaching this idea from a different direction, we shall obtain common characterizations of finite projective and affine spaces. Our principal result is the following.

Theorem 1. A finite incidence structure is isomorphic to the design of points and hyperplanes of a finite projective or affine space of dimension greater than or equal to4 if and only if there are positive integers v, k, and y, with μ > 1 and (μ – l)(v — k) ≠ (k — μ)2such that the following assumptions hold.

• (I) Every block is on k points, and every two intersecting blocks are on μ common points.

• (II) Given a point and two distinct blocks, there is a block containing both the point and the intersection of the blocks.

• (III) Given two distinct points p and q, there is a block on p but not on q.

• (IV) There are v points, and v– 2 ≧ k > μ.

In several papers, W. Klingenberg has elaborated the connections between Hjelmslev planes and a class of rings, called H-rings (4; 5; 6), which are rings of coordinates for the corresponding Hjelmslev planes. Certain homomorphic images of valuation rings are examples of H-rings. In these examples, the lattice of (right) ideals of the ring, say R,is a chain, and the coordinatization of the corresponding Hjelmslev plane yields a natural embedding of the plane in the lattice L(R3) of (right) submodules of the module R3. Now, L(R3) is a modular lattice with a homogeneous basis of order 3 given by the submodules a1 = (1, 0, 0)R, a2 = (0, 1, 0)R, a2 = (0, 0, 1)R, and the sublattices L(N, ai) of elements less than or equal to ai are chains. Forgetting about the ring, we find ourselves in the situation of a problem suggested by Skornyakov (7, Problem 23, p. 166), namely, to study modular lattices with a homogeneous basis of chains. Baer (2) and Inaba (3) investigated lattices of this kind with Desarguesian properties and assuming that the chains L(N, ai) were finite. Representations of the lattices by means of certain rings can be found in both articles.

In (1), Atkinson characterized the set of Fredholm operators on a Banach space X as those bounded operators invertible modulo the two-sided ideal of compact operators on X. It follows from this characterization that the Fredholm operators can also be described as those bounded operators which are invertible modulo the two-sided ideal of bounded operators on X which have finite-dimensional range. This ideal is the socle of the algebra of all bounded operators on X. Now, if A is any ring with no nilpotent left or right ideals, then the concept of socle makes sense (the socle of A in this case is the algebraic sum of the minimal left ideals of A, or 0 if A has no minimal left ideals). Also, in this case, the socle is a two-sided ideal of A. In this paper we study the elements in a ring A which are invertible modulo the socle. We call these elements the Fredholm elements of A.

Although the axiom of complete regularity ought to be a separation axiom, in none of its usual forms does it look like an intrinsic separation axiom. Our aim in this paper is to establish such characterizations of complete regularity which naturally fit in between regularity and normality and which already have proved to be fundamental and useful. This can simply be achieved by replacing the family of all open sets (as used in the formulation of the separation axioms) by some suitable (sub)base of open sets. For the sake of simplicity, we assume all our spaces to be T1 and we shall operate with (sub)bases of closed sets (instead of open sets). It is appropriate to define the notion of a screening.

Two subsets A and B of a set X are said to be screened by the pair (C, D) if C ∪ D = X, A ∩ D = ∅ and C ∩ B = 0. (Consequently, A ⊂ C and B ⊂ D.)

The problem of classifying the normal subgroups of the general linear group over a field was solved in the general case by Dieudonné (see 2 and 3). If we consider the problem over a ring, it is trivial to see that there will be more normal subgroups than in the field case. Klingenberg (4) has investigated the situation over a local ring and has shown that they are classified by certain congruence groups which are determined by the ideals in the ring.

Klingenberg's solution roughly goes as follows. To a given ideal , attach certain congruence groups and . Next, assign a certain ideal (called the order) to a given subgroup G. The main result states that if G is normal with order a, then ≧ G ≧ , that is, G satisfies the so-called ladder relation at ; conversely, if G satisfies the ladder relation at , then G is normal and has order .

Let be an associative algebra over the field and let be the group of all automorphisms and anti-automorphisms of which leave elementwise invariant. A function Fwith domain and range contained in is called an intrinsic functionon if (i) for each Ω in and (ii) F(ΩZ) = ΩF(Z) for every Z in .

Rinehart (5) has introduced and motivated the study of the class of intrinsic functions on , and has characterized these functions for the cases in which is the algebra of real quaternions, the algebra of n × ncomplex matrices, or the algebra of n× nreal matrices (5; 6). The algebras listed above, along with the algebra of n × nquaternion matrices, constitute the full list of possibilities for the simple direct summands of any semi-simple algebra over or ; see (2).

In (1, §§ 2.3 and 2.4) we proved that certain distributive lattices are simultaneously lattice-embeddable in the degrees of recursive unsolvability and in the hyperdegrees. Let ℒ be the non-distributive lattice {0,1, a0, a1,…}, where ai ∪ aj = 1 and ai ∩ aj = 1 whenever i ≠ j. We shall prove the following theorem.

THEOREM. The lattice ℒ is simultaneously lattice-embeddable in the degrees and hyperdegrees.

For A ⊆ N, let deg(A) and hyp(A) be the degree and hyperdegree of A, respectively. To prove the theorem we must construct hyperarithmetically incomparable sets A0, A1, … such that for Δ = deg, hypand for all distinct i, j:

1

2

Now, if each 〈Ai, Aj〉 were a generic pair in the sense of (1), then (2) would hold.

Hölder's theorem on archimedean groups states:

An ordered (abelian) group G is order isomorphic to an ordered subgroup of the ordered group R of real numbers if and only if it is archimedean.

We comprehend this theorem in the following setting: G is a Z-module and Ris the completion with respect to the open interval topology of the ordered field Q; Qitself is the ordered quotient field of the ordered domain Z.

Rephrasing the situation, we raise the following question: We start with a fully ordered domain A,let Kbe its ordered quotient field. We endow Kwith the open interval topology and consider , the topological completion of K. Is it possible to impose a compatible order structure on and if this can be done, when can we say that an ordered A-module Mis order isomorphic to an ordered A-submodule of ? In Theorem 3.1, we obtain a set of necessary and sufficient conditions for this isomorphism to hold.

In the present note we introduce a straightforward algebraic generalization of inner product spaces, which we appropriately name generalized inner product (GIP) spaces. In the same fashion in which different topologies :an be introduced in inner product spaces, adequate topologies can be introduced in GIP spaces in such a manner that topological vector spaces are obtained. We enumerate and derive some fundamental properties of different topologies in GIP spaces, having primarily in mind their possible later application to quantum physics.

The desirability of having in quantum physics more general structures than Hilbert spaces (in which quantum mechanics is usually formulated) is suggested by Dirac's formalism (2), which deals with “unnormalizable” vectors. Unfortunately, although this formalism is very elegant from the point of view of the facili ty of dealing with its symbolism, it completely lacks in mathematical rigour.

In (12), Montgomery and Samelson conjectured that an MS-fibering of polyhedra with total space an n-sphere must have a homology sphere as its singular set. Mahowald (11) has shown that, indeed, an orientable fibering with n ≧ 4 must have a Z2-cohomology sphere as its singular set, while Conner and Dyer (4) have shown this for n arbitrary provided the fiber itself is a Z2-cohomology sphere. We show that if the singular set is tame, then it is a Z-homology sphere if the fiber is also one. This result together with those of Stallings (15), Gluck (7), and Newman and Connell (13) are applied in the case where the singular sets are locally flat and tame. It is shown (Theorem 5.2) that MS-fiberings of spheres on spheres, with closed connected manifold fibers and singular sets, are topologically just suspensions of (Hopf) sphere bundles. In a subsequent publication, the case where the singular sets are finite shall be considered. The reader is invited to consult (3) and (18) in this case.

Let f: Mn→ Npbe the projection map of an MS-fibering of manifolds β with finite non-empty singular set Aand simply connected total space (see 1). Results of Timourian (10) imply that (n, p) = (4, 3), (8, 5) or (16, 9), while a theorem of Conner (2) yields that #(A), the cardinality of the singular set, is equal to the Euler characteristic of Mn. We give an elementary proof of this fact and, in addition, prove that #(A) is actually determined by bn/2(Mn), the middle betti number of Mn, or what is the same, by bn/2(Np – f(A)). It is then shown that β is topologically the suspension of a (Hopf) sphere bundle when Np is a sphere and bn/2(Mn) = 0. It follows as a corollary that β must also be a suspension when Mn is n/4-connected with vanishing bn/2. Examples where bn/2 is not zero are constructed and we state a couple of conjectures concerning the classification of such objects.

A famous theorem of Hardy asserts that if f ∊ H1, then the sequence of Fourier coefficients satisfies . For this reason we say that the sequence (1, 1/2, 1/3, …) belongs to the multiplier class (H1, l1). In this paper, we investigate the multiplier classes (Hp, l1) for 1 ≧ p ≧ ∞. Our observations are based on the fact that a sequence (λ(0), λ(l), …) belongs to (Hp, l1) independent of the arguments of its terms. We also show that (Hp, l1) may be thought of as the conjugate space of a certain Banach space.

1. Preliminaries.Lp denotes the space of complex-valued Lebesgue measurable functions f defined on the circle |z| = 1 such that

is finite.

In (3), Isbell proposed a stronger definition for the term “complete category” and obtained many nice theorems for the resulting notion of a completion. In particular, he showed (3, Theorem 3.20) that completions of small categories satisfy a strong normality condition.

In this paper we shall always use the term “complete” in the weaker sense of Freyd (1). (In (3), Isbell used the term “small-complete” for this weaker notion.) We shall prove that the completions, in the sense of Freyd, of small categories also enjoy the same normality condition, provided they admit at least one bicategory structure. (The complete categories in the sense of Isbell always admit bicategory structures; see the remark following Proposition 2.4.)

In what follows, we let mean that is a full subcategory of . Moreover, if , then means that each object of is equivalent to an object in .

Various conditions under which an upper semi-continuous (u.s.-c.) decomposition of E3 yields E3 as its decomposition space have been given by Armentrout (1; 2; 5), Bing (7; 8), Lambert (13), McAuley (14), Smythe (17), and Wardwell (18). If the projection of the non-degenerate elements is 0-dimensional in the decomposition space, then “shrinking” or “Condition B” (6) has proven particularly useful.

In this paper we shall investigate monotone u.s.-c. decompositions of a locally compact connected metric space M, where the projection of the nondegenerate elements is 0-dimensional. We show in Theorem 1 that each open covering of the non-degenerate elements of a 0-dimensional decomposition has a locally finite refinement.

In § 5, we use Theorem 1 to investigate the following question which is similar to one raised by Bing (11, p. 19): Let G, G′, and G″ be decompositions of M such that the non-degenerate elements of G are those of G′ together with those of G′.