Search

The purpose of this note is to present a new proof of a generalized form of Day's fixed point theorem. The proof we give is suggested by the work of T. Mitchell in his paper, Function algebras, means, and fixed points, [2]. The version of Day's theorem which we present here has not appeared explicitly in the literature before, and seems especially well suited for application to questions concerning fixed point properties of topological semigroups. 1. Preliminaries. We adopt the terminology and notation of [2] except where otherwise specified. New terminology will be introduced as needed. Let y be a convex compactum (compact convex set in a real locally convex linear topological space E), and let A( Y) denote the Banach space of all (real) continuous affine functions on Y under the supremum norm. Observe that A(Y) contains every function of the form h=f\Y + r where fe E* and r is real; thus A(Y) separates points of Y.

1. Denote by G the group of the equation upon which depends the determi. nation of the curves of order n 3 having simple contact at 1 n ( n -3 ) points with a given curve C of order n having no double points. The case in which n is odd was discussed in the former paper (Transactions, January, 1902) and G was shown to be a subgroup of the group defined by the invariants 43, 04, , , * *, the latter group being holoedrically isomorphic with the first hypoabelian grouip on 2p indices with coefficients taken modulo 2. For n even, G is contained in the group H defined by the invariants 04' 069 * with even subscripts. JORDAN has shown (Traite, pp. 229-242) that H is holoedrically isomorphic with the abelian linear group A on 2p indices with coefficients taken modulo 2. The object of the present paper is to establish the latter theorem by a short, elementary proof, which makes no use of the abstract substitutions [al, 1 ., p, p1] of JORDAN, and which exhibits explicitly the correspondence t between the substitutions of the isomorphic groups.

In the paper with the above title, which appeared in these T r a n s a c t i on s, vol. 2 (1901), p. 428, I obtained a number of general theorems concerning the zeros of functions of the form 02 y' ol y, where y is a solution of a hornogeneous linear differelntial equation of the second order. There s a further general theorem of the same character which I should have included in that paper if I had discovered it at the time, and which I now give as VII' below, it being a counterpart to VII in the earlier paper. I also give three special applications of this general result, numbered VIII', VIII", VIII"', since they are counterparts of VIII. A part of VIII' was obtained by Fite in the Annals of M athematics for June, 1917, (see the closing lines of his article) by another method; and it was this special result of Fite's that suggested to me the much more general Theorem VII'. What follows should be regarded as an addendum to the former paper, to be inserted on page 434 at the end of ? 2. All references are to that paper, to which the reader should turn for an explanation of the notation and a statement of the restrictions placed on the functions. VII'. If none of the six functions

Walsh's Theorem II states that if the roots of f (z) are in the circular region C, and if z is exterior to C, then ai lies in C. This is precisely the result obtained by Laguerre as given on page 59 of volume I of his collected works, if expressed in non-homogeneous coordinates as on page 57. We shall use another form of Laguerre's Theorem (loc. cit., p. 57) to prove the underlying proposition on which Walsh bases his proof of his Theorem I, as follows: Every circle through any point z and its "derived point" a as defined by (1) either passes through all the roots off ( z ), or else has at least one root in the region interior to it, and at least one root in the exterior region. I have recently called attentiont to the fact that this is a corollary of a theorem of Bocher's on jacobians which has served as a starting point in Walsh's earlier papers. With a slight change in Walsh's notation, whereby we substitute z's for a's, the proposition from which Theorem I is deduced may be stated thus

This paper contains a number of disturbing misprints: Equation (2) p. 560 should read Gpf(x) = aoxPfm+ a1xPf11 + *** + amixpf + amx. Line 17 p. 561 read Ap(x) XBp(x) instead of Ap(x)Bp(x). The term perfect (volkommen) in Theorem 1 is used in the sense of Steinitz, Algebraische Theorie der Korper, edited by Hasse and Baer, pp. 50-51. Line 21 p. 562 should read Fp(x) = Qp(x) X (xP ax) + Ax. Equation (9) p. 562 should read A = aoa(Pm-l)/(P-1) + aia(P'1)/(P-1) + + am,2a0P+1 + am-,a + am. In the expression line 9 p. 563 the last term should be A(")x. Equation (17) p. 564 should read F'"(x) = Fnl(x)P -Fn-1(Wn)PFn-1(X). Line 8 from below p. 574 should read

Let C denote the category of compact Hausdorff spaces and H : C → H C H:C \to HC be the homotopy functor. Let S : C → S C S:C \to SC be the functor of shape in the sense of Holsztyński for the projection functor H. Let X be a continuum and H n ( X ) {H^n}(X) denote n-dimensional Čech cohomology with integer coefficients. Let A x = char H 1 ( X ) {A_x} = {\text {char}}\;{H^1}(X) be the character group of H 1 ( X ) {H^1}(X) considering H 1 ( X ) {H^1}(X) as a discrete group. In this paper it is shown that there is a shape morphism F ∈ Mor S C ( X , A X ) F \in {\text {Mor}_{SC}}(X,{A_X}) such that F ∗ : H 1 ( A X ) → H 1 ( X ) {F^\ast }:{H^1}({A_X}) \to {H^1}(X) is an isomorphism. It follows from the results of a previous paper by the author that there is a continuous mapping f : X → A X f:X \to {A_X} such that S ( f ) = F S(f) = F and thus that f ∗ : H 1 ( A X ) → H 1 ( X ) {f^\ast }:{H^1}({A_X}) \to {H^1}(X) is an isomorphism. This result is applied to show that if X is locally connected, then H 1 ( X ) {H^1}(X) has property L. Examples are given to show that X may be locally connected and H n ( X ) {H^n}(X) not have property L for n > 1 n > 1 . The result is also applied to compact connected topological groups. In the last section of the paper it is shown that if X is compact and movable, then for every integer n, H n ( X ) / Tor H n ( X ) {H^n}(X)/{\operatorname {Tor}}\;{H^n}(X) has property L. This result allows us to construct peano continua which are nonmovable. An example is given to show that H n ( X ) {H^n}(X) itself may not have property L even if X is a finite-dimensional movable continuum.

This paper continues the study of the smoothness properties of (real) topological linear spaces. First, the smoothness results previously obtained about various important classes of locally convex spaces, such as Schwartz spaces, are improved. Then, following the ideas of Bonic and Frampton, we use these results to give sufficient conditions for the existence of smooth partitions of unity on manifolds modelled on topological linear spaces. 1. Preliminaries. In order to make this paper self-contained we include the definition of the two types of derivative used and the definition of an 8-category. We will employ the definitions of the derivative in topological linear spaces investigated in detail by Averbukh and Smolyanov [2], [31. See also [14], [15] and [191. Let TLS denote the class of all Hausdorff topological linear spaces over the real field R. Let 2 1(E, F) = 2(E, F) denote the set of all continuous linear maps from E into F, where E, F e TLS. We define by induction p(E, F) = 2(E, p 1(E, F)). Each ?1p(E, F) is given the topology of uniform convergence on bounded subsets of E. If X is a topological space, W(X) will denote the class of all open subsets of X. Let f: U -' V, where U e ((E), V E C(F), E e TLS and F c TLS. Then we say f is Fre6cbet differentiable at x e U, if there exists u e S(E, F) such that for each bounded subset B of E and for each o-neighbourhood W in F, there exists 8 > 0 such that f(x + th) f(x) u * th E tW, whenever b E B and ItL

This paper deals with the construction of somep-groups of maximal class. Introduction. Let G be a group, G2 = [G, G] be the commutator subgroup of G and G,,I = [Gi, G] for i > 2. A p-group G of order pT is said to be of maximal class if IG: G21 = p2 and IGi: Gi+II = p for i = 1, . . ., n 1. In addition if G is of maximal class then GI is defined to be the largest subgroup of G such that [GI, G2] 2p, w 5 for the groups under consideration are known when p = 2 or p = 3. When p = 2, GI is cyclic, so G is metacyclic; when p = 3, G2 is abelian, that is G is metabelian, so these groups are also easy to classify [1, p. 82]. The structure of the groups is quite different when p > 5. If G is of order pf and G. is the first abelian member of the lower central series for G then, as we shall see, w may be as large as n/4. The other assumptions, G, is of class 2 and n > 2p, are made to keep commutator calculations from getting out of hand. The following conventions, assumptions, and results will be used throughout this paper: G is a p-group of maximal class and order pf where n > p + 2; GI, G2, ... are defined as above; w is the smallest positive integer such that GW is abelian; G is generated by s and s, where s E G G, and si E G -G2; si = [s1_1, s] for i = 2, 3, .. ., n; G, = for i = 1, 2, . . ., n -1; S = 1 for m > n. The symbol a(i, v) will always denote an integer such that 0 3. Those groups where w 3 then [G,-,, GJ] < G,-p+3. This was proved by Blackburn [1, p. 77]; it will also be proved here for it follows quite easily from results we have and need for other purposes. The relation [G,-,, GJ] < Gn_p+3 is equivalent to p-4 [Swl, W J~-II 5a(wqlNq q=O Received by the editors July 22, 1971. AMS (MOS) subject classifications (1970). Primary 20D15; Secondary 20F05.

In this paper we generalize a result of Tamura on .S-indecomposable semigroups. Based on this, the concept of a minimal sequence between two points, and from a point to another, is introduced. The relationship between two minimal sequences between the same points is studied. The rank of a semigroup S is defined to be the supremum of the lengths of the minimal sequences between points in S. The semirank of a semigroup S is defined to be the supremum of the lengths of the minimal sequences from a point to another in S. Rank and semirank are further studied. Introduction. Semilattice decompositions of semigroups were first defined and studied by Clifford [1]. Since then several people have worked on this topic, notably Tamura (5H{9]. The author's work on the subject can be found in (3], (4]. In this paper, we start by generalizing a result of Tamura (8] (or (9]) on Sindecomposable semigroups. Based on this, the concept of a minimal sequence between two points, and from a point to another, is introduced. The relationship between two minimal sequences between the same points is studied. The rank of a semigroup is defined to be the supremum of the lengths of the minimal sequences between points in the semigroup. The semirank of a semigroup is defined to be the supremum of the lengths of the minimal sequences from a point to another in the semigroup. Rank and semirank are further studied. To understand this paper, the reader need only be aware of the first few chapters of Clifford and Preston [2] and Tamura's decomposition theorem. (See any of (5], [6], [8] or (9]. It was rediscovered by Petrich [(0].) I Preliminaries. Throughout, S will denote a semigroup and Z+ the set of positive integers. A congruence a on S is called a semilattice congruence if S/a is a semilattice. S x S is the universal congruence on S. S is S-indecomposable if S x S is the only semilattice congruence on S. Definition. Let a, b E S. Then (1) a I b if and only if b E SaS1. I is transitive and reflexive. (2) is defined as a > b iff a I bi for some i E Z'; let -0 denote --, i.e., = _-. (3) a nI b iff there exists x E S such that a x b. (4)a biffa bforsome Z+. (5) -is defined as a b iff a a; let -? denote -, i.e., -? Received by the editors May 9, 1972 AMS (MOS) subject classfications (1970). Primary 20M10.

Let T be any finite 2-group which has a normal four-group but has no normal Abelian subgroup of rank 3, and assume T is not the dihedral group of order 8. If T is a Sylow 2-subgroup of a finite fusion-simple group G, it follows (Thompson) from Glauberman’s Z ∗ {Z^ \ast } -theorem that T has exactly one normal four-group, say W. This paper establishes what isomorphism types of T can so occur under the hypothesis that N G ( T ) = T C G ( T ) {{\mathbf {N}}_G}(T) = T{{\mathbf {C}}_G}(T) and the three nonidentity elements of W are not all G-conjugate. All T arrived at in this paper are known to so occur. The reason for this hypothesis is that the similar situation for T with a normal four-group and no normal Abelian subgroup of rank 3, where T is a Sylow 2-subgroup of a finite simple group G but without the above hypothesis, had been analyzed earlier by the author (under her maiden name, MacWilliams; Trans. Amer. Math. Soc. 150 (1970), 345-408).

The concept of isotopy plays an extremely important role in the structure theory of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals. We take Koecher's characterization of isotopy and use it as the basis of a definition of semi-isotopy. It is clear that semi-isotopies induce, in a natural way, automorphisms of the lattice of inner ideals. We concern ourselves with the converse problem; namely, if 71 is a semilinear bijection of a quadratic Jordan algebra such that 71 induces an automorphism of the lattice of inner ideals, is 71 necessarily a semiisotopy? We answer the above question in the affirmative for a large class of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals (said class includes all such algebras of capacity at least three over fields of characteristic unequal to two). Moreover, we prove that the only such maps which induce the identity automorphism on the lattice are the scalar multiplications. 1. Preliminaries. In this paper, we continue our study of automorphisms of the lattice of inner ideals of simple quadratic Jordan algebras begun in our earlier paper [7]. In that paper, attention was focused on determining the automorphism group of the lattice of inner ideals and on determining under what conditions two algebras can have isomorphic lattices of inner ideals. In the present paper our attention is focused on determining which semilinear bijections of a simple quadratic Jordan algebra satisfying the minimum condition on principal inner ideals induce automorphisms of the lattice of inner ideals. We will freely make use of our earlier results and of McCrimmon's determination of the inner ideals for the algebras under consideration [9]. The concept of an isotope of a quadratic Jordan algebra plays an extremely important role in the structure theory (see [3] ). One defines an isotopy to be an isomorphism of a quadratic Jordan algebra onto an isotope. Received by the editors July 20, 1973. AMS (MOS) subject classifications (1970). Primary 17C20, 17C30; Secondary 16A68.

This paper presents some new theorems concerning recursively enumerable (r.e.) sets. The aim of the paper is to advance the search for a decision procedure for the elementary theory of r.e. sets. More precisely, an effective method is sought for deciding whether or not an arbitrary sentence formulated in the lower predicate calculus with sole relative symbol c is true of the r.e. sets. The main achievement of the paper is the characterisation of the hh-simple sets as those coinfinite r.e. sets whose r.e. supersets form a Boolean algebra. The reader is referred to Davis's book [1] for basic information about the partial recursive (p.r.) functions and about r.e. sets. Other background material required for a proper understanding of the present paper consists of [8], [3, Theorem 2], [10, Introduction and ?4], and [5] where the contributions have been listed in their natural order. We take the formulation of the lower predicate calculus given in Abraham Robinson [9]. Natural numbers are denoted by lower case Roman letters and sets of them by lower case Greek letters. The empty set is denoted by 0 and the set of all natural numbers by v. The complement of any set a is denoted by a'; a is called cofinite or coinfinite just if a' is finite or infinite respectively. For sets a, / we write aoc3 just if the set (a -3 u (3 a) is finite; otherwise we write aZ 3. By function we mean a map of some subset of v x v x **. x v into v; functions will be denoted by upper case Roman letters as will relations on the natural numbers. The informal logical signs used are v , &, ->, , (x), (Ex), -,> which are to be read as " or ", " and ", " implies ", "not", "for all x", "there exists x", "is equivalent to" respectively. Let ,' be a finite class of propositions; then s otherwise sup a is to be oo. The plan of the paper is as follows. In the first section we give a brief discussion of the elementary theory of r.e. sets and prove that its decision problem is of the same degree as that of the elementary theory of the lattice obtained by taking the equivalence classes of r.e. sets with respect to -. In ?2 we prove the main theorem which states: if a is an r.e. subset of an r.e. set / then either there exists a recursive subset 8 of:/ such that a u 8 = 3 or there exists a recursive sequence {8il of disjoint finite subsets of / such that 8i a is nonempty for all i. This theorem was inspired by

The " universe of discourse " of the present paper is the class of all welldefined systems (K, R) where K is any class of elements A, B, C, ..., and R is any triadic relation. The notation R [ABC], or simply ABC, indicates that three given elements A, B, C, in the order stated, satisfy the relation R. Examples of such systems (K, R) are the following, of which example (a) is the most important: (a) K is the class of points on a line; AXB means that the point X lies between the points A and B. (b) K is the class of natural numbers; AXB means that the number X is the product of the numbers A and B. (c) K is the class of human beings; AXB means that X is a descendant of A and an ancestor of B. (d) K is the class of points on the circumference of a circle; AXB means that the arc A-X-B is less than 180O. (e) K is a class comprising four elements, namely, the numbers 2, 6, 6, an 6 A X Ce A 1V1) 74= A .T) anu O?o; LLa n mieans~ A =1 At D. It is obvious that these systems, and others like them, will possess a great variety of properties expressible in terms of the fundamental variables K and R. The object of this paper is to state clearly the characteristic properties of the type of system represented by example (a) above, by which this type of svstem is distinguished from all other possible systems (K, RI). In Section 1, we give a basic list of twelve postulates, due essentially to Pasch,t from which various sets of independent postulates will later be selected.

Introduction. The first part of the paper gives a class of polyhedral 3manifolds in the 4-sphere S4 and with homology groups H(M) —H2(M) =0, but with 7ri(M) 5^0. Thus some "Poincare" spaces are imbeddable in Si. Also, necessary and sufficient conditions are given for a group to be Hi(M) for some 3-manifold M in S4. In Part Two a negative answer is found for the following question: Since the metric cases of 2-gcms reduce to the classical types [12, IX], and the 3-gcms generally do not, can one state that a spherelike 3-gcm in S4 must be a classical manifold? We construct 3-gcms in Si which have the same homology and homotopy character as the 3-sphere (globally and locally), but which are not locally euclidean. In part three of the paper a negative answer is found for a question proposed by Griffiths [6] by showing the existence of a 3-dimensional "homotopy manifold" which is not locally euclidean. This is a space previously obtained by R. H. Bing by identifying the points on a certain set of tame arcs in S3. Part I. Let P be a connected 2-polyhedron which is contained in S4 as a subcomplex of some simplicial decomposition 2 of Si. Before anything else is done we make a barycentric subdivision of 2. This is to make sure that the resulting decomposition of P will be complete in the sense that if all of the vertices of a simplex of .S4 lie in P, then the simplex itself is in P. Now every simplex of St(P) — P is the join of a simplex of P with a simplex in Si — St(P). Hence each xE(St(P)—P) may be assigned a number ^(x) between 0 and 1, its "distance from P". Newman [8] has shown that the set M= [x|/(x) = l/2] is a manifold. The set M is clearly 3-dimensional, poly

In a paper with the above title published in these Transactions* there appeared a statement that "B(x) becomes infinite as x approaches x". In a letter, dated October 18, 1921, Professor Hans Hahn called attention to the fact that the proof of this statement, as implied by the context, although not explicitly mentioned, was insufficient, because it depended upon the nonvanishing of the quadratic form ,5 jij Cj cj, which had been proved to be definite by the aid of the fact that Z(x) + 0, and this condition does not hold at x1. It is the purpose of this note to supply a proof of the statement quoted above, upon which the validity of the concluding theorem of the earlier paper depends. I acknowledge my indebtedness to Professor Hahn for having pointed out the need for this supplemenitary proof. Starting with the definition of B(x) as given on page 435 and putting rij (x) = ikP (xI) Z(kj) (x), we have B (x) = j rij cj cj/Z. We shall show first that not every rij can vanish at x1 of the same order as Z(x). For, suppose that Z(x) = (x-xl)s Z1 (x), Z, (x1) + 0. If now every ij (x) had the factor (xx)s, then, since IRk((Xi)l Ot it would follow that every Z(Jk) (x) had it also. From the definition of the functions Z(Jk) (x), we derive the following system of equations

Presented to the Society August 23, 1956; received by the editors November 29, 1956. (1) Research sponsored by the Office of Ordnance Research, U. S. Army, under Contract No. DA-11-022-ORD-2196. (2) Gauss [3] considered the functions F(a, b, c, z) and called them "contiguous" if the values of one of the first three elements differed by unity. Heine [6, p. 102] called them averwandte Reihe." (3) Numbers in brackets refer to the bibliography at the end of the paper. (4) Professor Perron [7] uses eu instead of q, which Heine used, for the reason that qa, for example, is not single valued. However, by the use of eua, which is single valued, one does not need to specify the branch of qa. However, since this paper was written, Professor Perron's third edition, volume II, of Die Lehre von den Kettenbraichen, has been published (Teubner, Stuttgart, 1957). In this new edition Professor Perron again uses q instead of eu (cf. p. 125, formula (19)).

1. The present paper is a continuation of a previous paper on the subject of relatively uniform convergence of sequences of functions of a single real variable.t The results of both piapers admit of immediate extension to more general domains. A sequence {f, (x) } of functions, defined on an interval I = (a x b), converges relatively uniformly to a limit function f (x) if there exists a function o ( x ), called a scale function o( x ) such that for every small positive number e there is an integer ne such that for every n greater than ne the inequality If (x) fn (x) I e -e(x)I

and the above results also follow from the classical theory of continued fractions (for example, see [10] and the remarks in [5]). In view of this, it seems natural to attempt a study of the properties of the polynomials, Pn(x), as determined by the coefficients, cn and Xn. One approach to such an investigation is of course through the study of the convergence of the continued fraction (1.3). In this paper, however, we shall avoid direct reference to the theory of continued fractions although we shall make fairly extensive use of the results from the problem of moments. Instead, our investigation centers upon the "true" interval of orthogonality of Pn(x) }. [By the "true" interval of orthogonality, we mean, following Shohat (see [5; 6, p. 113]), the smallest interval which contains in its interior all zeros of all Pn(x).] In this connection, the "chain sequences" of Wall [10] arise naturally and play a fundamental role. Thus our study is largely "arithmetical" in character. Throughout this paper, "orthogonal polynomials" means polynomials

is any sequence of rational integers satisfying (1.1), then after a certain point the sequence becomes periodic when considered modulo m. Its least period is called the characteristic number of the sequence (U) modulo m. In a recent paper in these Transactionst, I have considered the problem of determining this characteristic number given m, C, C2, , * * Ck and the k initial values UO, U1, *, Uk-i of the sequence (U), and I have reduced it to certain basic problems in the theory of higher congruences.$ In the present paper, I am concerned with the following problem which I shall similarly reduce to a problem in the theory of higher congruences: Given any positive integer s: to find the number of distinct sequences (U) modulo m whose characteristic number is exactly equal to s. 2. I obtain here the following results.

in compactly generated atomic lattices was developed which for the most part generalized the classical theory for finite dimensional lattices. It was shown that if a compactly generated atomic lattice is semimodular, then every element has an irredundant meet decomposition into completely irreducible elements. Every element of a compactly generated atomic lattice has a unique irredundant decomposition if and only if the lattice is locally distributive. Aside from uniqueness the most fundamental arithmetical property of irredundant decompositions is the Kurosh-Ore replacement property, that is, for every pair of irredundant decompositions a n Q = nlQ' of an element a and for each irreducible qC-Q there is an irreducible q'EGQ' such that a=q= nn(Q-q). Every modular lattice has this property. Moreover, it was shown that a semimodular, compactly generated, atomic lattice has the replacement property if and only if the lattice is locally modular. All of these results used heavily the semimodular condition. Only the characterization of unique decompositions was carried out for general lattices, and this was accomplished because semimodularity could be proved in this case. The present paper extends the decomposition theory to include nonsemimodular lattices. It is shown in the second section that the elements of any compactly generated atomic lattice have irredundant decompositions. In the third section a necessary and sufficient condition is obtained for an arbitrary compactly generated atomic lattice to have the replacement property. This condition is a modification of the lower-semimodular law, and in the presence of semimodularity is easily seen to be equivalent to local modularity. Throughout this paper the notation and terminology of [I ] is used. Lattice join, meet, inclusion, proper inclusion, and covering, are denoted respectively by the symbols U, C, _

Then the function f(z) has both a rational multiplication and a rational addition theorem. We propose to determine all such functions. Our problem is a generalization of one treated in a paper by Ritti1), in which all periodic meromorphic functions having a multiplication theorem were determined. This is equivalent to taking S(z) =z in (2). For this case, Ritt proved that \m\ ^ 1 ; and if \m\ > 1, then f(z) is a linear function of one of the functions eaz, cos(az+s), p(z+s); when gs = 0, g»2(z+j3); when g2 = 0, S>'(z+s) and j?3(z+j3). Here a is arbitrary while s is restricted to certain values. If \m\ =1, then it was shown that m is either —1 or a third, fourth or sixth root of unity. The forms for the function f(z) were also explicitly given in this case. Use will be made of the results quoted and also of the methods of Ritt's paper. We distinguish the following three cases : (A) m >1; (B) m ?1, with m not a rational root of unity; (C) m a primitive wth root of unity. For Case (A), which is the one usually considered in multiplication theorems, we can restate the problem in a way which shows its connection with the theory of a class of functions first systematically studied by Poincare(2). He provided an existence theorem for meromorphic functions satisfying (1) assuming that 7?(z) has a fixed point(3) a for which 7?'(a) has a modulus greater than unity. Thus our problem is equivalent to that of finding all Poincare functions having a rational addition theorem.

During the past twenty years extensive work has been done on the ring C(X). The pioneer papers in the subject are [8] for compact X and [3] for arbitrary X. A significant part of this work has recently been summarized in the book [2]. Concerning the ring C(X, Z), very little has been written. This is natural, since C(X, Z) is less important in problems of topology and analysis than C(X). Nevertheless, for some problems of topology, analysis and algebra, C(X, Z) is a useful tool. Moreover, a comparison of the theories of C(X) and C(X, Z) should illuminate those aspects of the theory of C(X) which derive from the special properties of the field of real numbers. For these reasons it seems worthwhile to devote some attention to C(X, Z). The paper is divided into six sections. The first of these treats topological questions. An analogue of the Stone-Cech compactification is developed and studied. In ?2, the ideals in C(X, Z) are related to the filters in a certain lattice of sets. The correspondence is similar to that which exists between the ideals of C(X) and the filters in the lattice of zero sets of continuous functions on X. This theory provides a characterization of those ideals of C(X, Z) which are intersections of maximal ideals. ?3 is concerned with the space of maximal ideals in C(X, Z). In ?4, some existence theorems for maximal ideals are proved. The residue class fields of C(X, Z) modulo maximal ideals are studied in the last two sections. It turns out that those of prime characteristic are trivial: the integers modulo the characteristic. The residue class fields of characteristic zero are distinctly nontrivial. In ?5, the cardinality of such fields is investigated. The main result is that they are always uncountable. In ?6, the algebraic properties of the zero characteristic residue class fields are examined. It is shown for example that these fields are always quasialgebraically closed. As we noted above, very little has been published concerning the ring C(X, Z). Nevertheless, a considerable number of "folk theorems" exist in the subject. One of our objectives in writing this paper is to get these results

By a central topological group we mean a group G such that G/Z is compact, where Z (or Z(G)) denotes the center of G. A locally compact group satisfying this condition will be called a [Z]-group, and [Z] denotes the class of these groups. In [7] we developed the structure theory of [Z]-groups; in particular, we showed that every [Z]-group G obeys the following Structure Theorem: G = V x H (direct product), where V is a vector group and H contains a compact open normal subgroup K(3). Both the structure theory and, as will be shown in this paper, the representation theory of [Z]-groups generalize and unify in a natural manner the corresponding theories for compact groups on the one hand and for locally compact abelian groups on the other. In fact, many of the features common to these two classes appear in their natural setting only when viewed as being characteristic of [Z]-groups. In addition, there are strong indications that [Z] marks the utmost degree of generality in which all these features are still present; not the least of these is the fact that the representation theory of [Z]-groups is essentially finite-dimensional. By contrast, that of the slightly larger class of [FIA]--groups is not; an [FIA]-group being a locally compact group G such that Z(G), the group of inner automorphisms, has compact closure in %(G), the group of all topological group automorphisms of G, in the natural topology. The class [FIA] was introduced by R. Godement [4]. (For a full discussion of the relation between [Z] and [FIA] -, see [7].) The finite-dimensionality of representations referred to above in combination with the compactness condition which defines [Z] allows us to obtain less general but considerably sharper results than those laid down by Godement in [6]. The present paper gives complete proofs of the results announced in Bull. Amer. Math. Soc. 72 (1966), 836-841, under the same title. The paper is organized as follows. After establishing the basic definitions and a number of technical results in ?1 we proceed to the proof of the fundamental fact that continuous irreducible unitary Hilbert space representations of [Z]-groups are finite-dimensional (Theorem 2.1) and we obtain an orthogonality relation for the

Free projective planes n? (generated by n points on a line and two points off that line) have been studied for some time. Until now, however, virtually nothing was known about the collineation groups of the planes itn except that the groups possessed no central collineations. In this paper, the collineation groups will be studied. We shall show that the collineation group of ir2 has three generators and exhibit them. Then, the orbit under this group of those points which lie on generating quadrilaterals will be determined. Finally, a subgroup H" of the collineation group of 7r" (n > 2) will be defined and it will be shown that there exists an integer m such that for every n > 2, Hn is generated by at most m collineations. It is the author's conjecture that Hn is in fact the full collineation group of itn, but the methods of the present paper do not seem to generalize to give this result.

The object of this paper is to study the groups formed by the residue classes of a certain type of Kronecker modular system and some closely related generalizations of well-known theorems in number theory. The type of modular system to be studied is of the form 9) = (mn, mn-1, , Mnl, m). Here m, defined by (ml, m2, ***, Mk), is an ideal in the algebraic domain Q of degree k. Each term mi, i = 1, 2, *.., n, belongs to the domain of integrity of Qi = ( Q, xl, x2, * , xi), and is defined by the fundamental system ((1t/, t'2j), ***, i,t')). The various 1(i), j = 1, 2, *.., jI, are rational integral functions of xi with coefficients that are in turn rational integral functions of xl, x2, ... , xj_i, with coefficients that are algebraic integers in Q . In every case the coefficient of the highest power of xi in each of the 't') shall be equal to 1. We shall see later that the developments of this paper also apply to modular systems where the last restriction here cited is omitted, being replaced by another admitting more systems, these new systems in every case being equivalent to a system in the standard form as here defined. Any expression that fulfills all of the conditions placed upon each {() with the possible exception of the last one, we shall call a polynomial, and no other expression shall be so designated. This definition includes all of the algebraic integers of Q. Throughout this paper we shall deal exclusively with polynomials as here defined. The first part of this paper will contain the introduction with the necessary definitions and a discussion concerning the factoring of the system 9). The second section will then be devoted to setting up necessary and sufficient conditions that a set of residue classes belonging to 9) form a group when taken modulo 91. In the third section we shall study the structure of such a group with respect to groups belonging to certain modular factors of 9), besides

Introduction. The problem of constructing all the finite metabelian groups (that is, groups with an abelian commutator subgroup) is fundamentally settled by the Schreier theory of group extensions. In fact, a metabelian group may be considered as the extension of an abelian group 2a by an abelian group a; this is the simplest nontrivial case of a group extension. According to Schreier's theory, an arbitrary group 5 which is the extension of 2 by a is obtained by the following procedure: First, we have to find an automorphism group of 2f which is the homomorphic image of a, that is, if ais the automorphism corresponding to a Ea, then (AB) =A Ba, (A)a =Aa 5=A " for every A, B GX, o, rEG. Secondly, we have to find a factor system Ca, in A, satisfying CaT, CaT= C7,P for every pi o, ir (see [18, p. 90])(l). If Sa is a symbol denoting a certain representative of oin 0, then the relations SaAS,i =Aa, S,S, = C?,7Sa7 uniquely determine an abstract group 5 with the required properties. Unfortunately, the general formulation of the Schreier theory does not indicate (except in the most trivial cases) how to determine and specify the automorphisms and factor systems so that each 5 shall be obtained in one and only one way. The invariant characterization of solvable groups, even in the relatively simple case of metabelian groups, still remains one of the most important and most difficult problems of abstract group theory. At the present, it seems that the problem can be successfully approached only if we impose certain restrictions upon the family of groups to be determined. In a recent paper [12] I have determined all the groups (5 which have an abelian invariant subgroup 21 of the type (p, * *, p) such that 9/2 be cyclic. In the present paper a more extensive class of metabelian groups will be determined and completely characterized by numerical invariants. Whereas no restriction will be imposed upon the structure of the abelian invariant subgroup X, it is assumed that 9/f is cyclic and its order n is not divisible by the square of any prime number which divides the order of 2W. In particular, the latter condition is fulfilled if either n is squarefree, or if n and the order of 2a are relatively prime. Among the more important cases included in the above category (others will be mentioned in part 3) perhaps the most notable is the case of p-groups

0. Introduction and notation. In this paper we study the condition that the maximal right quotient (MRQ) ring Q [10, p. 106] of a right nonsingular ring R with 1 is flat as a left R-module. It is known [11, p. 134] that if Q is the classical right quotient ring of R, then Q is flat as a left R-module. This is not always the case with the MRQ ring of R: in ?2 we obtain an ideal theoretic characterization (Theorem 2.1) and a module theoretic characterization (Theorem 2.2) of a right nonsingular ring R, all of whose regular right quotient rings are flat as left R-modules; we also indicate the existence of a class of commutative rings R, whose singular ideal is zero and for which the maximal quotient ring is not R-flat. Throughout this paper R denotes an associative ring with identity. A right R-module M is denoted MR; all R-modules are unitary. Let NR and MR be modules such that NR. (MR. We say that NR is large in MR (MR is an essential extension of NR) if NR intersects nontrivially every nonzero submodule of MR. A right ideal I or R is large in R if IR is large in RR. For any module MR, L(MR) denotes the lattice of large submodules of MR. Let MR be a module. We denote by Z(MR) the singular submodule of MR. If for any x E M we set (0: x) ={r E R I xr =0}, then

1. Introduction. Finite non-Desarguesian projective planes have been known for all orders pf, where p is prime, n _ 2, and pn_ 9, except when p = 2 and n is a prime ? 5. In this paper a new class of projective planes is defined, having the orders 2n where n > 5 is not a power of two, thus establishing, in particular, the existence of non-Desarguesian planes of the missing orders. The new planes are coordinatized by semifields (sometines called division algebras, non-associative division rings, or distributive quasifields), which are algebraic systems satisfying the axioms for a field except with a loop replacing the multiplicative group. It is easy to show that finite proper semifields, i.e., semifields which are not fields, must have the orders pn where p is prime, n _ 3, and pn > 16. Proper semifields of these orders have been known to exist except as above, when p = 2 and n is prime; therefore the new systems show that proper semifields do exist for any order not excluded by simple arguments. A detailed treatment of the general theory of semifields and their relation to projective planes may be found in [3]. The manner in which these planes where discovered is perhaps as interesting as the planes themselves, since computers played a key role in the discovery. The author had received copies of two tables prepared by R. J. Walker (see [4]), which listed all commutative semifields of order 32 for which, if x is a generating element, x(x(x(x2))) = x + 1 or x2 + 1, respectively. There were 24 solutions in each table, and so it seemed plausible that a rule could be found yielding a correspondence between one set of solutions -and the other. A few hours of "cryptanalysis" did, in fact, result in the discovery of such a rule; and, since one of the solutions for the table x(x(x(x2))) = x2 + 1 was the field GF(32), the corresponding system for the other table could be written in a simple algebraic form [2]. After this, there was little difficulty showing that the same construction could be generalized to the construction of proper semifields of all orders 22k+1 with k > 1, and subsequent generalizations yielded the systems described in this paper. Therefore, we have an example in which

DEFINITION. A continuous curve every subcontinuum of which is a continuous curve will be called a perfect continuous curve.t In this paper, among other things, a study is made of this type of continuous curve. An important problem in analysis situs is to determine whether a given point set is arc-wise connected. It is known that a connected subset of a perfect continuous curve is not necessarily arc-wise connected.: Here, however, it is shown in Theorem 5 that, if a and b are two points of a connected subset N of a perfect continuous curve, then the set common to N and the arcs ab of N' is connected. And in Theorem 7 it is shown that if N' contains but a countable number of arcs ab then N contains an arc ab. ? A generalization of the problem as to whether there exists an arc joining two points in a given point set is the following: when do there exist in a given point set where q is a given positive integer, q arcs, distinct except for end points, joining two given points; or still a further generalization would be to ask when do there exist q such arcs joining two distinct closed point sets? This problem is considered in ?II of this paper. It might also be asked when do there exist q distinct arcs joining two distinct closed point sets? This problem is considered in ?I. I wish to express my thanks and to acknowledge my great indebtedness to Professor R. L. Wilder for his suggestions, criticisms, and constant encouragement.*

* Presented to the Society, under a slightly different title, December 29, 1902. Received for publication February 2, 1903. [For an abstract of an unpublished paper on the same subject, presented to the Society by the writer on April 26, 1902, see Bulletin of the American Mathematical Society, vol. 8 (1901-02), p. 371.] tCf. D. HILBERT, Ueber den Zahlbegriff, Jahresbericht der deutscben Mathematiker-Vereinigung, vol. 8 (1900), pp. 180-184. -The axioms for real numbers enumerated by HILBERT in this note include many redundancies, and no attempt is made to prove the uniqueness of the system which they define. (Cf. Theorem Il below.) The main interest of the paper lies in his new Axiom der Vollstiindigkeit, which, together with the axiom of Archimedes, replaces the usual axiom of continuity. The other axioms are given also in his Grundlagen der Geometrie (1899), ? 13. Complete sets of postulates for particular classes of real quantities (positive integral, all integral, positive real, positive rational) can be found in the following papers: G. PEANO, Sul coneeito di numero, Rivista di Matematica, vol. 1 (1891), pp. 87-102, 256-267; Formulaire de Mathematiques, vol.3 (1901), pp. 39-44.-Here onlythepositive integers, or the positive integers with zero, are considered. An account of these postulates is given in the Bulletin of the American Mathematical Society, vol. 9 (1902-03), pp. 41-46. They were first published in a short Latin monograph by PEANO, entitled Arithtmetices principia nova mnethodo e.xposita, Turin (1889). A. PADOA: 1) Essai d'une theorie algebrique des nombres entiers, pr-ecede d'une introduction loypque Aunetheorie deductive quelconque, Bibliotheque du congres international de philosop h i e, Paris, 1900, vol. 3 (published in 1901 ), pp. 309-365; 2) Numeri interi relativi, R i v i s ta di Matematica, vol. 7 (1901), pp. 73-84; 3) Un nouveau systeme irreductible de postulats poutr I'algebre, Compte rendu du deuxiieme congr6s international des math6mati ciens, Paris, 1900 (published in 1902), pp. 249-256.-The second of these papers is an ideographical translation of the first; the third reproduces the principal results. E. V. HUNTINGTON: 1) A complete set of postulatees for the theory of absolute continuous maignitude; 2) Complete sets of postulats for the theories of positive integral and positive rational numbers; Transactions, vol. 3 (1902), pp. 264-279, 280-284.-The first of these papers will be cited below under the title: Magnitudes. [ln the fifth line line of postulate 5, p. 267, the reader is requested to change "one and only one element A " to: at least one element A -a typographical correction which does not involve any further alteration in the paper.] Among the other works which may be consulted in this connection are: H. B. FINE, The numsber system of algebra treated theoretically and historically, Boston (1891). 0. STOLZ und J. A. GMEINER, Theoretische Arithmetik, Leipzig (1901-02). Two sections of this work have now appeared. G. PEANO, Aritmetica generale e algebra elementare, Turin (pp. vii + 144, 1902). 358

Introduction. The purpose of this paper is to clarify the structure of finite groups satisfying the following condition: (CN): the centralizer of any nonidentity element is nilpotent. Throughout this investigation we consider only groups of finite order. A group is called a (P)-group if it satisfies a group theoretical property (P). In this paper we shall clarify the structure of nonsolvable (CN)-groups and classify them as far as possible. This goal has been attained in a sense which we shall explain later. If we replace in (CN) the assumption of nilpotency by being abelian we get a stronger condition (CA). The structure of (CA)-groups has been known. In fact after an initial attempt by K. A. Fowler in his thesis [8], Wall and the author have shown that a nonsolvable (CA)-group of even order is isomorphic with LF(2, q) for some q=2n>2. A few years later the author [12] has succeeded in proving a particular case of Burnside's conjecture for (CA)groups, namely a nonsolvable (CA)-group has an even order. Quite recently Feit, M. Hall and Thompson [7] have proved the Burnside's conjecture for (CN)-groups. We can therefore consider groups of even order and focus our attention to the centralizers of involutions. We consider the condition (CIT): (CIT): a group is of even order and the centralizer of any involution is a 2-group. There is no apparent connection between the class of (CN)-groups and the class of (CIT)-groups. But a nonsolvable (CN)-group is a (CIT)-group (Theorem 4 in Part I). This theorem reduces the study of nonsolvable (CN)groups to that of (CIT)-groups. Both properties (CN) and (CIT) are obviously hereditary to subgroups (provided that we consider only subgroups of even order in the case of (CIT)). Although it is true that a homomorphic image of a (CN)-group is also a (CN)-group (this statement is false for infinite groups), it is not an obvious statement. On the other hand it is not difficult to show that a factor group of a (CIT)-group is a (CIT)-group, provided that the order is even. This is due to the following characterization of (CIT)-groups: namely a (CIT)-group is a group of even order containing no element of order 2p with p>2 and vice versa. This makes the study of (CIT)-groups somewhat easier. The large part of this paper concerns the structure of (CIT)groups.

In this paper we develop algebraic and geometric conditions which imply that a given proper Dehn map can be replaced by an embedding. The embedding, whose existence is implied by our theorem, retains most of the algebraic and geometric properties required in the original proper Dehn map. I. Preliminaries. In this paper all spaces will be simplicial complexes and all maps will be piecewise linear. The boundary, closure, and interior of a subset X of a space Y will be denoted by bd (X), cl (X) and int (X), respectively. A proper map f, taking a space X into a space Y, is a map such that f bd (X) = bd (Y) n f(X). A Dehn map f, taking a compact, bounded, two-manifold F into a 3-manifold M, is a map such that (1) bd (F)=f-lfbd (F); (2) fIbd (F) is a homeomorphism. All unlabeled homomorphisms are natural maps induced by inclusion. We shall consider the problem of replacing a proper Dehn map of a compact, connected, orientable, bounded surface F into a compact, (necessarily) bounded, irreducible, orientable 3-manifold M with an embedding. As will be pointed out later, the requirement of orientability may not be strictly necessary. The author would like to thank J. Gross for a number of helpful conversations. In [3] Papakyriakopoulos proved Dehn's lemma, that is, a proper Dehn map f of a disk 9 into a 3-manifold M can be replaced by a proper embedding g of 9 into M such that g bd (9) =f bd (9). A natural question to consider is the following: Let f: F -* M be a proper Dehn map of a compact, connected, bounded, orientable surface F into a compact 3-manifold M. Does there always exist a proper embedding g: F1 -? M such that (1) g(bd (F1))'f(bd (F)); (2) genus (F1)

Introduction. We begin with a definition. Let X be a topological space and let e(X) be the set of all continuous functions which map X into R, the real numbers. Let J = {f1, ,f...4J be a set of n distinct functions in W(X) such that each nonzero linear combination of these functions over R has at most n 1 zeros in X. It is clear that this property of F is equivalent to the property that given any set of n distinct points {x1,-.,xn} in X, the matrix ||fi(xj) || has an inverse. We may call such a set b5 a real-valued unisolvent system of order n on X, or for brevity, an n-R-U system on X(1). For examples, confer (1.6) and (2.5) below. The purpose of this paper is to characterize topologically those spaces which admit n-R-U systems. It should be noted that any nonempty topological space of less than n points has an n-R-U system and that any nonempty topological space whatsover has a 1-R-U system. Consequently in this paper we shall consider only spaces with at least n points, where n > 2. A summary of this problem's background is useful. For this the following definition is needed. Let X4 be an n-dimensional vector R-subspace of ro(X), where X is a locally compact Hausdorff space and TO(X) is the vector R-subspace of W(X) containing all functions which "vanish at infinity" (i.e., all functions f in W(X) such that for each ? > 0, the set {x E X: If(x) j >, } is compact). X' is said to be a Haar subspace of TO(X) if given anyf in TO(X) there is a unique gf in X such that

In [1] Bing constructed a 1-dimensional hereditarily indecomposable continuum which is the inverse limit of a sequence of circles Ci and such that the mapsfi: Cj-*Cjj were of degree 1. In this paper we show that the dimension cannot be raised, i.e., if S =Lim(SN, fi) where SiN is an N-sphere and ft is essential, then S is not hereditarily indecomposable. In doing so we further generalize Bing's definition of "e-crooked." Lemma 2 shows how to construct hereditarily indecomposable continua by making the bonding maps sufficiently crooked. In Lemma 3 we get a necessary crookedness condition on the boinding maps if the limit space is to be hereditarily indecomposable. The remainder of the paper is devoted to proving that in the case of N-spheres (N> 1) this condition cannot be satisfied. DEFINITIONS AND NOTATION. Let Xi be a sequence of compact metric spaces, and for i >2 let fi be a map of Xi into Xi-1. Then the subspace (2) (3)

1. The normalizers of the system normalizers are subgroups of some importance in the theory of solvable groups initiated by P. Hall [1-5]. For example, P. Hall observed [4] that a system normalizer was contained in the hypercenter of its norm lizer. R. Carter analyzed extensively the class of solvable groups which have self-normalizing system normalizers [6]. Furthermore, Carter, in a recent paper [7], has pointed out the importance of the question of the homomorphic invariance of the normalizers of the system normalizers. That is, if H is the normalizer of a system normalizer D of a solvable group G and N is a normal subgroup of G, is HN/N the normalizer of the system normalizer DN/N of G/N? This is equivalent with the following question(2): If DN/N is the normalizer of the system T of G/N, is there a system S of G with normalizer D such that S is carried onto T by the natural homomorphism of G onto G/N? For these reasons we have undertaken an investigation of the subgroups in the title of this paper. Our first result gives some more information about the embedding of a system normalizer in its normalizer.

In a recent paper by the same title [II], hereafter referred to as Part I, R. H. Cameron and the present author investigated the structure of the class of additive functionals of class L2 on the Wiener space C [VI] and obtained certain other results concerning functionals additive and merely measurable on C. Recently Cameron conjectured that every functional additive and measurable on C was of class L2(C), and hence that the characterization and representation theory previously developed applied to all functionals additive and measurable on C. It is the purpose of this paper to establish Cameron's conjecture. Unfortunately the proof is not easy and depends on a number of preliminary results. However, the techniques of proof are novel enough to be of some interest. In addition, the results permit considerable strengthening of our earlier theorems and a curious characterization of the class of continuous linear functionals on the Hilbert space of real-valued functions of class L2 on the unit interval. We begin by introducing a number of preliminary concepts and theorems. DEFINITION 1. Let {I'pk} be a C. 0. N. set (complete orthonormal set in the L2 sense) on [0, 1], each of whose elements is of bounded variation, and let f(t)x E L2 on [0, 1]. Let fjt) be the nth partial sum of the orthogonal development of f(t) in terms of {I ik ; that is, let fn(t) = EL' Ckkp A(t), where Ck = If (t)k(t) dt. Then we define the P. W. Z. (Paley-Wiener-Zygmund) integral f f(t) dx(t) by the equation ff(t) dx(t) = lim ffn(t) dx(t), x E C O ~~~~~n -o 00 whenever the limit on the right exists. It has been shown by Paley, Wiener, and Zygmund [IV] that the limit defining the P. W. Z. integral exists for almost all x E C and that this limit is essentially independent of the particular choice of the C. 0. N. set {Ipk I in the sense that if IOk} and {lk* } are any two C. 0. N. sets all of whose elements are of B. V., then 1 1 (, f f(t) dx(t) = (O*) ] f(t) dx(t) for almost all x E C. It has also been o ~ ~~~~ I shown in [IV] that f fn(t) dx(t) converges in the L2(C) mean to a f(t) ax(t) that the P. W. Z. integral agrees with the ordinary Riemann-Stieltjes integral almost everywhere on C when f(t) is of B. V., and that if G(u1, **, un) is

Introduction. This paper generalizes certain methods of decomposition of compact metric continua due to R. L. Moore [3; 4] and G. T. Whyburn [9; 10; 13]. While their methods yield acyclic continuous curves, hyperspaces are obtained here which are aposyndetic [1 ] continua. The concept of a con. tinuum being aposyndetic is a generalization of the concept of continuous curves and was introduced in 1941 by F. B. Jones. In compact metric continua, this idea is equivalent to Whyburn's notion of semi-locally-connectedness [14; 2]. A continuum M, i.e., a closed and connected point set, is said to be aposyndetic at a point p with respect to a point x provided that there exists a subcontinuum N of M and an open subset 0 of M such that M-xDNDODp. If M is aposyndetic at a point p with respect to each point x of M-p, then M is said to be aposyndetic at p. It is said that M is aposyndetic if M is aposyndetic at each of its points. In an early paper [10], Whyburn made use of connected cuttings of a compact metric continuum M to obtain a decomposition of M into an acyclic continuous curve. Later, he made use of nonseparated cuttings [13] to obtain a decomposition of a continuous curve into a nondegenerate acyclic continuous curve. Certain of these theorems concerning nonseparated cuttings are generalized in obtaining an aposyndetic decomposition. Moore [3] has obtained decomposition theorems by use of certain sets M(P) defined as follows: For each point P of a compact continuum M, let M(P) denote the set of all points X of M such that there do not exist uncountably many different points each separating P from X in M. He proved the following theorem: If M is a compact metric continuum and G is the collection of all point sets M(P) for all points P of M, then G is an upper semi-continuous collection of disjoint continua filling up M and G is an acyclic continuous curve with respect to its elements as points. The definition of the sets M(P) due to Moore may be generalized in the following manner: Suppose that M is a continuum and that p is a point of M.

which occurs in all theories of numerical functions hitherto considered. The two most highly developed theories of this kind are those in which multiplication in the ring of all numerical functions is abstractly identical with C (Cauchy) or D (Dirichlet) multiplication of infinite series.t Lehmer's five postulates are sufficient for the development of a theory of inversion as exemplified in the cases of C, D multiplication, without requiring, as is the fact in those cases, that the function ?(x, y) (his {t(x, y)) of composition be a polynomial. For C multiplication, q (x, y) _x+y -1, instead of the usual x+y, as a change in notation justifies; for D multiplication, 0(x, y) _xy. But these are not the only ?(x, y) which give an arithmetical theory of composition (as in the papers cited); to mention only three further instances, there is von Sterneck's "L.C.M. calculus," quoted by Lehmer, in addition to the well known compositions ?(x, y) M(x, y), where M is either "max" or "min." It is of considerable interest then to see precisely what position is occupied in the general theory of composition developed in the paper (B) by the classical theories in which multiplication is abstractly identical with either C or D. We shall prove that if (p(x, y) is a polynomial in x, y, then, in order that the composition + (x, y) = n, where n is an arbitrary constant integer >0 and x, y are variable integers >0, shall lead to an arithmetical theory of composition, it is necessary and sufficient that + (x, y) be either x +y 1 or xy, namely, that multiplication in the ring of all numerical functions be either C or D.

This paper is a continuation of an earlier paper(2) of the same title. The semigroups described in the title are called threads. Part I determined the structure of all possible threads with a zero element. The present paper will determine all threads without zero. The numbering of the sections, lemmas, and theorems will continue that of Part I. References to the literature in square brackets will be to the bibliography of Part I. Let S be a thread. Since S is connected and has endpoints, it is compact. Bv a theorem of Numakura(3) and Wallace [11, Lemma 4], S containis a kernel K, i.e. an ideal contained in every ideal of S. K is closed and connected (Wallace [11, Lemma 4]), and so must be a closed subinterval [a, w] of S. K degenerates to a single point (a =cw) if and only if S has a zero; since this case was handled in Part I, we assume throughout Part II that a

1. Introduction. In [2], we proved that if G is a cellular decomposition of a 3-manifold M such that the associated decomposition space is a 3-manifold N, then M and N are homeomorphic. In this paper we shall establish a related result for 3-manifolds with boundary. We shall show that if G is a cellular decomposition of a 3-manifold with boundary M such that the associated decomposition space is a 3-manifold with boundary N, then M and N are homeomorphic. The techniques of this paper have applications to the study of embeddings of curves and surfaces in 3-manifolds. Suppose that M i's a 3-manifold with boundary and G is a cellular decomposition of M such that the7 associated decomposition space is a 3-manifold with boundary N. By Theorem 2 of this paper, M anld N are homeomorphic. Suppose K is a surface or a curve in M such that no nondegenerate element of G intersects K. If P denotes the projection map from M onto N, then PIK is a homeomorphism. It is natural to ask the following: Is P[K]'embedded in N the same way that K is embedded in M? In ?5, we give an affirmative answer to this question. In particular, K is tame if and only if P[KJ is tame. In ?6 we shall show that if G is a cellular decomposition of a 3-manifold with boundary into a 3-manifold with boundary, then, the projection' map can be approximated arbitrarily closely by homeomorphisms. In [2]; we established a theorem of basic importance in the study of cellular decompositions of 3-manifoids that yield 3-manifolds as their decomposition spaces. The main result, Theorem 1, of this paper is a useful corollary of the results of [2]. The results mentioned in the preceding two paragraphs are applications of Theorem 1. In [4], we shall apply Theorem 1 to the study of shrinkability conditions which are satisfied by certain cellular decompositio'ns of E3 that yield El as their decomposition space. 2. Terminology and notation. The statement that M is a 3-manifold with boundary means that M is a separable metric space such that each point of M has a neighborhood in M which is a 3-cell. If M is a 3-manifold with boundary, a point p of M is an interior point of M if and only if p has an open neighborhood in M which is an open 3-cell. The interior of M, mnt M, is the set of all boundary points

Introduction. The theory of regressive isols was introduced by J. C. E. Dekker in [7]. The results that we wish to present in this paper belong to this theory and is a continuation of some of our studies in [1], [3] and [4]. We will assume that the reader is familiar with the terminology and some of the main results of the papers listed as references. We let E denote the collection of all nonnegative integers (numbers), A the collection of all isols, A* the collection of all isolic integers, and AR the collection of all regressive isols. Iff is a function from a subset of E into E then 8f will denote its domain and pf its range. Let un and vn be two one-to-one functions from E into E. Then un_ * vn, if there is a partial recursive function f such that

Let G be a locally compact group with a left Haar measure 'p'. Let 'p' and 'r' be two real numbers > 1. By Lrp conjecture we mean the following assertion: whenever f e Lr(G) and g E LP(G) we have that the convolution product f * g of f and g is defined and belongs to LP(G) again if and only if G is compact. By LP conjecture we mean the assertion above with r =p. Both the conjectures were widely believed to be true though there was no written statement about these conjectures until recently. But in 1960, Kunze and Stein [3] showed that the Lr,p conjecture is false for the unimodular group of 2 x 2 real matrices. This naturally raises the question whether the LP-conjecture is true in general. The first published result on the LP-conjecture is by Zelazko [9] and Urbanik [7] in 1961. They proved the conjecture to be true for the abelian case. Then in [5] the author established the truth of the LP-conjecture for discrete groups when p ? 2. The author announced in that paper that the conjecture is true for all groups when p > 2 and presented this result to Amer. Math. Soc. in August of 1963 at the Boulder meeting. At the same time Zelazko [10] established the conjecture for allp > 2 for all unimodular groups. He claims to have established the conjecture for p > 2 for all groups in that paper but his crucial Lemma 1 of that paper contains a gap in the proof. In a private communication, Zelazko agreed to this gap. The problem is still open in general when p > 1. In this paper we prove the following: The LP-conjecture is true for all locally compact groups when p > 2. The LP-conjecture is true for totally disconnected groups when p =2. The methods used in this paper yield the truth of the conjecture for all nilpotent groups, and all unimodular C-groups of Iwasawa when p > 1. But this result will appear elsewhere. This is a portion of the author's doctoral dissertation submitted to Yale University in 1963 under the guidance of Professor C. E. Rickart. The author had a grant AFOSR 62-20 from the Air Force when this work was done. The author thanks Professor C. Ionescu-Tulcea for having brought the paper [9] of Zelazko to his attention. Notations and conventions. All purely topological notions are taken from [2]. All topological spaces occurring in this paper are taken to be Hausdorff. All notions in topological groups and integration on locally compact groups are

This paper is concerned with certain aspects of acyclicity in a compact connected topological semigroup, and applications to the admissibility of certain multiplications on continua. The principal result asserts that if S is a semigroup on a continuum, finitely floored in dimension 2, then S=ESE implies S=K. Introduction. We are concerned here with some aspects of acyclicity in a compact connected semigroup, and applications to the admissibility of a semigroup structure on "finitely floored" spaces (Definition 2). The main result is that if S is a semigroup on a continuum, finitely floored in dimension 2, then S=ESE implies S= K. This is a generalization of the result of Cohen and Koch [1], where the same conclusion is obtained assuming that S is a floor for each nonzero h E H2(S), cd S_ 2, and S is locally Euclidean except possibly at one point. A preliminary result which may be of interest in itself is that if S is a semigroup with a zero satisfying S =ESE on a continuum, then for each nonzero h E H2(S) there exists a pair of idempotents e andf such that hiSe u Sf# O. The notation is that of [4]. In particular, S denotes a topological semigroup, K the minimal ideal, and E the set of idempotents. For a closed set A of S, S/A denotes the space obtained by shrinking A to a point. The cohomology used is that of Alexander-Wallace-Spanier with coefficient group arbitrary. Throughout the paper we shall use reduced groups in dimension 0. We denote by A\B the complement of B in A, the closure of A by A* and the empty set by 1i. If Hn(A). The following theorem is due to Wallace [5]: THEOREM 1. Let X and Y be compact Hausdorff spaces and < a closed relation from X to Y such that L(A) r) L(B) is connectedfor each pair of closed subsets A and Received by the editors June 29, 1970. AMS 1970 subject classifications. Primary 22A15; Secondary 22A15.

For some time now, very little has been known about abelian subgroups of pgroups.We shall try and remedy this situation in a series of papers, of which this is the first. One impetus for doing this was created by several natural conjectures which arose in problems in other areas of group theory. In this paper we shall study p-groups with respect to the existence and nonexistence of abelian subgroups which in one sense or another can be considered as large subgroups. Later work will deal with several other topics. Our first result is in a negative direction; we shall demonstrate the falsity of the conjecture that every group of order pn has an abelian subgroup of order pn/2. Previously, we announced [1] that the best one could hope for was abelian subgroups of order at least p*48n, but for odd primes we can greatly improve this result by an entirely different method.

This paper reports some recent results in the theory of generalized parametric surfaces. The object of this theory, introduced by Young in [18], is to generalize the elementary notions of parametric surface in such a way that a general principle of minimum holds for surface integral variational problems in parametric form. Since nonregular problems are not excluded, the notion of generalized parametric surface is of necessity more general than the usual concepts of surface, involving in many instances both a track (surface in the ordinary sense, possibly mildly discontinuous) and a linear average over a space of Jacobian matrices at each point of the track (for ordinary surfaces the average is the trivial one in which the normal has weight one and all others weight zero) (2). In this paper, we first introduce abstract notions of boundary and track, defined for all non-negative linear functionals L on a certain linear space. Since, in particular, every generalized parametric surface is such a linear functional, these definitions apply to all generalized parametric surfaces. We next specialize to admit only boundaries which correspond in a certain way to boundary curves in the sense of classical analysis, and characterize the set of all L with boundary in this class. Finally, we specialize further by imposing a condition of indecomposibility with respect to the boundary. As an application, some conditions are established for equivalence of indefinite variation problems which have fixed boundary conditions with positive definite problems. 1. Introduction. The main definitions and results of the paper areoutlined in this section. Details and auxiliary results are left to later sections. Let Em (mr> 3) denote the space of all continuous functions f(x, J) of an m-dimensional vector x= [xi, . . ., xm] and an mXm skew-symmetric matrix J = [jr8] of rank 2 which satisfy the homogeneity condition f(x, kJ) = kf(x, J) for k _ O. A mapping x(u, v) from the square U: [O < u < 1; 0 < v

These are actually the Pringsheim inequalities applied to the even and odd parts of f [2, p. 161 ]. The rp are nonnegative numbers (depending, in general, upon the cp), and inequality is required in the first two in case the cp are different from zero. Leighton [1] had employed these inequalities in the case rp = I cp I0, and had found thatf converges if, in addition, lim sup I cpl, for p = oo, is finite. Scott and Wall showed that if the r, are subjected to certain restrictions, for example, r,=1, or lim inf rlr2 r . *, = 0, then the divergence of the series Z bp , where b1 = 1, cp= 1/b bp+?, p= 1, 2, 3, * * * (the series to be counted as divergent if some cp vanishes), is necessary and sufficient for the convergence of f. In this paper we show that the same conclusion holds without any restruction upon the rp (?4). This is accomplished by showing, more generally, that when the even and odd parts of a continued fraction f are absolutely convergent (?2), then f converges if, and only if, the series I bp| is divergent. We base our investigations upon the linear fractional transformation w = Tp(z) which carries the points oo, 0, and 1 into fp_l fp, and fp+, respectively, where fo=O0, f=1, f2=1/(1+c1), * * is the sequence of approximants of f. Other results in this paper include: a theorem connecting the fixed points of T, with the convergence and value of f (?2), sufficient conditions for absolute convergence of f (??3, 6), and some new convergence regions for f (?5). The latter are found by means of certain geometrical properties of the inequalities (1.1). 2. The transformation Tp. It will be convenient to introduce the following notations: D is the set of "points" c = (ci, C2, C3, ) such that the denominators Bo=1, B1=1, Bp+1=Bp+cpBp_i, p= 1, 2, 3, * * , of f=f(c) are all different from zero

In this paper we define a partial ordering on a class of topological spaces as follows('): For two spaces X and Y in the class, we will say X Y which is an onto a-map. (Recall that f is an a-map if there is an open cover ,B of Y such that f-'(13) refines a.) It is easy to check that the relation < is reflexive and transitive. In [7], Kuratowski proved that the closed n-dimensional ball is not _ the n-sphere. More recently, Ulam [8] raised the question whether the 2-dimensional ball is < the 2-dimensional torus. Ulam's question has stimulated interest in the ordering itself, and a result of T. Ganea [5] states, roughly speaking, that if X < Y and Y is a compact n-dimensional manifold and X is an absolute neighborhood retract, then X has the same homotopy type as an n-dimensional manifold. This result then gives Ulam's question a negative answer(2), and suggests studying properties of a space Y which are inherited by a space X satisfying X _ Y. The purpose of this paper is to present certain properties which are inherited, in the above sense. Also, comparison is made between this ordering and the ordering given by dimension (Theorem 1), and (for the class of absolute neighborhood retracts) the ordering given by a-domination for every cover a (Theorem 2). Since the property of possessing a fixed point under every continuous function will be shown to be inherited, another question of Ulam's is answered here [8, Chapter IV, Problem 12].

Cubic varieties of 003 poillts in space of four dimensions have been extensively studied by CASTELNUOVO t and by SEGRE, t showing the close connection between the section of the enveloping cone by ordinary space (apparent contour) and the well knowll congruences of the second and third orders, with their focal surfaces. The treatment is exclusively synthetic and the methods there employed do not lend themselves readily to the discussion of particular cases. In the following paper I wish to call attention to two particular projections of the variety having nine double points. This variety is given a three page discussion in SEGRE'S first paper, containing most of the results of Articles 1, 2 3, 4, 6 of the present paper. The result of Art. 5 is stated by CASTELNUOVO in a footnote without proof, but no use is there made of it. The results in the remaining articles are new. I have made extensive use of all the papers cited, but as I shall proceed analytically, the development will be along diflerellt lines, and no knowledge of their contents will be assumed. @ Let r 1Z2z3 + \z4z5z6 ° define a cubic variety in space of four dimensions 4, where Ewt O. § Every plane defined by wi ° ( i 1, 2, 3 ) and wk-O (k 4, 5, 6 ) will lie entirely on r. These nine planes may be designated by av. Any two of these having neither subscript in common will intersect in just one point which is a double point on r. Since two values of i and of k have been used to define each point, it ulay be denoted by A with the remaining pair as subscripts. It follows immediately that r contains nine double ?oints; through every double point pass four planes and ill every plane lie four double points.

Introduction. A Banach algebra in this paper will be understood to mean a commutative, semi-simple Banach algebra with multiplicative unit e. By the carrier space DA of the Banach algebra A, we mean the space of multiplicative linear functionals on A to C, the complex numbers, and it is to be endowed with the usual weak* topology (cf. [6]). For a E A, a denotes the Gelfand transform of a defined on (DA and A will denote the collection of such functions. We set the following notation. x will be used to denote an indeterminate over A as well as over A and C. If x(x) = 7=Oa,ix' is a polynomial over A, let &(x) and ;,(x) denote, respectively, ,i oQX,(2) = 0}, ac(x) A[x], plays an important role in the present paper. Z(oc(x),A) is topologized with the relative product topology from (A x C. The mapping n is defined by r(h, 2) = h, (h, 2) e Z(oc(x), A). The multiplicity function M of a(x) is defined as follows: for (h, 2) E Z(x(x), A), M(h, A) is equal to the multiplicity of A as a root of cth(x) = 0. In ?1 we introduce the concept of M-neighborhood of a point in Z(a(x),A). We say that W c Z(cx(x), A) is a M-neighborhood of (ho, 2A) E Z(oc(x), A) if W is a neighborhood in Z(oc(x),A) of (ho,AO) and if, for each h e p(W), M(ho, 2O) is equal to the sum of the values of M at the points (h,2) in 7K-1(h) nl W. Proposition 1.1 states that M-neighborhoods exist and that they form a base for the neighborhood system at each point of Z(ox(x),A). The remainder of this section contains most of the topological lemmas needed for our work on factorization. Particular attention is paid to the case where Z(o(x),A) contains a compact open subset K (7t(K) = 'DA) on which M is constant. When this condition obtains, K and 4FD decompose topologically and this decomposition in turn forces a(x) to factor. The main factorization theorem (2.1) says that if a(x)eA[x] and if K (nr(K)-=(A) is a compact open subset of Z(a(x), A), then there exists a monic polynomial P(x) e A[x] such that ,B(x) is a factor of o(x), Z(fl(x), A) = K and Z(ax(x)/fl(x), A) = Z(a(x), A) K. A more detailed description of the factorization of monic polynomials follows. For example, it is shown that if A is indecomposable

A compact continuum W is said [1](1) to be a simple web if there exists an upper semi-continuous collection G of mutually exclusive continua filling up W and another such collection H also filling up W such that (1) G is a dendron with respect to its elements and so is H and (2) if g and h are elements of G and H, respectively, the common part of g and h exists and is totally disconnected. Hence, a simple web is a web(2). Examples of simple plane webs. Although a simple web is not necessarily a subset of the plane, this paper will deal only with those of this type. A square plus its interior is a simple web. We may consider the elements of G to be intervals parallel to one pair of sides of the square and the elements of H to be intervals parallel to the other pair of sides. We see from Theorem 1 of this paper that if C1, C2, * * * are circles no two of which intersect each other and such that C1 incloses Ci (i = 2, 3, * * . ), then C1 plus its interior minus the sum of the interiors of C2, C3, * is a simple web. A square plus its interior plus an interval intersecting both the interior and the exterior of the square is not a simple web. Hence, a plane web is not necessarily a simple web. By the use of Theorem 1 of this paper, we find that the continuous curve described on page 273 of [7] which is left connected but not locally connected on the omission of some countable subset is not a simple web. This paper gives necessary and sufficient conditions that a compact plane continuum be a simple plane web. Professor R. L. Moore suggested the problem of finding such conditions. Much credit is due him for the development of this paper. It has been shown [6] that every simple plane web is a continuous curve. In the present paper, the following theorem will be established.