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In [4], Phelps showed that in certain function algebras the unit ball is the closed convex hull of its extreme points. The algebra, C(X), of complex valued continuous functions on a compact Hausdorff space, will always have this property. The class of logmodular algebras which have a Gleason part which is total also was shown to have the property. In this paper we give an elementary proof of the first result (a proof which is, in theory at least, constructive). The simplest nontrivial example of a logmodular algebra with a total part is the disk algebra (i.e. the functions continuous on the closed disk and analytic in the interior). For this algebra we show that the extreme points (in fact the exposed points) of the unit ball form a dense subset of the boundary of the unit ball. Let U be the unit ball of C(X). It is well known that q in U is an extreme point of U if I q(x) I = 1 for all x in X. Now if f in U never vanishes, then f is in the closed convex hull of the extreme points; in factf is between two uniquely determined extreme points. We need only observe that for each x in X,f(x) is halfway between two uniquely determined extreme points of the disk and that these points vary continuously with x. Now it is not necessarily the case that each function in U can be approximated by a nonvanishing function. For a counterexample we need only look at h(z) = z in the algebra of continuous functions on the disk. (If If(z)-zj 1/e. Letfi, i = 1, *, N be the retract maps obtained from these rotations. Then I 1/N Efi(z)-z

This note deals with the correction of a small flaw in the proof of a theorem (and a lemma) in [2]. Also a simpler proof of the theorem in question is given. In the proof of Theorem 2.3 in [2], the statement "that distinct L's cannot belong to a common pure subgroup HL follows immediately from the fact that no two L's generate with G [p] the same subgroup of G" is false as it stands. One has to exercise a little more care in the selection of HL. For example, since the union [xa, Xa]aEA of all the L's (in Lemma 2.1 in [2]) is still linearly independent, there exists according to [1] a pure subgroup H of G supported by S such that HDL for each L. Hence one could choose HL=H for each L. However, all goes well if HL is chosen such that HLn [Xa, XTg aEA =L, as can easily be done. Similar remarks hold for Lemma 3.8 in [2]. A short alternate proof of Theorem 2.3 in [2] is given. In what follows let c be the cardinality of the continuum and Q be the first ordinal whose cardinality is c. The notation and terminology will be the same as that in [2].

M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.

F(k) ={xkj x F}; F,=F; Fk= (Fk_l,F). Let (u, v, 0)=u, (u, v, 1)=(u, v), (u, v, n)=((u, v, n-1), v). Then Sanov [3] proved that (1. 1) (u, v, apa _1) p-a E F(p#)Fapa+?, /3 a = 1, 2, where p is a prime. In this paper, (1.1) is proved for the cases a= 1, 2; 3 arbitrary. A slight generalization of these results is also proved. Sanov's proof involved an investigation of ideals in a Lie Ring. In this paper, Hall's Collection Process will be used. The method also yields other formulas, e.g. (1.2) (u, v, p2 1)P' (E F2p2_pF(p) (1.3) (u, v, pa+l 1)p~l C F2pa+l paF(pl), a = 1 2, and can be used to produce numerous formulas of a similar nature. The author hopes that some of these formulas and/or the method may be of use in solving other group-theoretic problems. The author was unable to use the method to prove (1.1) for a =3. Note that for a=f = 1, (1. 1) becomes (1.4) (U, v, p 1) E F(p)Fv+1. (1.4) plays an important role in the theory of the Restricted Burnside Problem.

If BP denotes the space of smooth alternating p-forms on the C' manifold M, we are interested in finding the spaces of R-linear maps from iPP to be that commute with the diffeomorphisms of M. For a compact manifold M these spaces were found by R. S. Palais. In this note we find them for noncompact M. Let V, W be the spaces of sections of two tensor bundles over a connected Coo manifold M of dimension n. In a paper [3] appearing in Trans. Amer. Math. Soc. 92 (1959), R. S. Palais studied a number of the spaces 4 (V, W) of the R-linear maps V->W natural with respect to the action on V and W of the group G of diffeomorphisms on M. We sketch the definition of this action: If g-G we define R,:F(TM) -4F(TM) to be the differential of g, and Rg:r(T*M)-->F(T*M) to be the dual of the differential of g-1. For the other spaces of tensor fields Rg is defined as the obvious tensor product of these two R-linear isomorphisms. A map c: V->W is then called natural if cRg =R,c for all g G. Palais was particularly interested in the case when V and W are the space IP of p-forms and the space bq of q-forms respectively, 0 < p, q

The first part of the theorem of the above paper states that if ois a closed convex set in real X1 ... Xn-space which is invariant under permutations of coordinates, and if C(o-) denotes the set of real symmetric nXn matrices whose eigenvalues X1, X. form the coordinates of points in a, then C(o) is convex. I am obliged to Professor R. T. Rockafellar for pointing out that this statement is essentially contained in a theorem of Chandler Davis.2 The statement also follows from an earlier result of V. B. Lidskii,3 which was not known to me at the time of publication.

C. Hobby and C. R. B. Wright [2] have just published the following Theorem A. However, their proof seems to contain an error.2 The notation of [2] is used except that G. is not reserved for the nth term of the lower central series of G: +/(G) denotes the Frattini subgroup of G; (G, H) means the group generated by the commutators g-lh-1gh where gEG, hEH; (A1, A2, * * *, A.+,) is defined inductively as ((A1, A2, . , A.), A.+,); HCG means that H is properly included in G.

PROOF. By a theorem of Rudin a function gEH' in U whose boundary values are real a.e. on I can be analytically continued to D [3, p. 59]. The lemma follows on applying Rudin's result to gi= (1/2) (fl+f2) and g2=(i/2) (fi-f2). PROOF OF THEOREM 1. By a well-known decomposition theorem [2, p. 87], f(z)=B(z)F2(Z), where B(z) is a Blaschke product and F(z) EH1. Since the boundary values of B (z) have absolute value one a.e. on K, we have a.e. on I, f(ei0)= |f(eio)I, or B(ei0)F2(ei0) = F2(ei0) |, and hence

A p-ring (or generalized Boolean ring) P is a ring of fixed prime characteristic p in which ax=a for all a in P. In this paper P is partially ordered by a relation which is a generalization of the usual Boolean order. A subset S of P is then called quasiorthogonal if ab(a-b)=O for all a, b in S. It is shown that P is injective in the category of p-rings if and only if every quasiorthogonal subset has a supremum under this partial order. Sikorski [4] has shown that in the category 9 of Boolean rings the injective objects are the complete Boolean rings. The purpose of this paper is to present a generalization of this result to the category 9 of p-rings, where 9 is understood to be the category with objects rings P of fixed prime characteristic p in which aP=a for all a e P and with morphisms the usual ring homomorphisms. If P is a p-ring, then the set B(P) of idempotents of P is a Boolean ring under the multiplication of P and the new addition defined by aE3b=a+ b-2ab. Batbedat [2] has used B to establish an isomorphism between b? and R. (See also Stringall [6].) Hence, the injective objects in 9 are simply those P for which B(P) is a complete Boolean ring. In this paper the irnjective objects in b? will be characterized by a kind of completeness of a particular partial order that is an extension of the usual partial order on the Boolean ring of idempotents. DEFINITION ]. For a, b E P, a_b if and only if aP-1b=a. It is easily shown that _ is a partial order on P (see, e.g., Abian [1]), but in general (P, _) is not a lattice. It is, however, a lower semilattice with meet defined by aAb=a-a(a-b)P-1. A simple calculation shows that a(bAc)=abAac and aAO=O for a, b, c e P. Foster [3] has shown that every element a of P is uniquely expressible as a sum of elements {ej(a): jE Z.} of B(P). In particular, v-1 (1) a = Zjej(a) i=l Presented to the Society, January 26, 1973 under the title Injectivity in the category of p-rings; received by the editors February 27, 1973. AMS (MOS) subject classifications (1970). Primary 06A70; Secondary 06A40, 06A23.

Results connecting binary quadratic forms and their associated quadratic fields are extended to irreducible decomposable forms and their associated fields. A rational linear substitution that carries such a form into a nonzero rational multiple of itself is shown to correspond with a linear map which admits a unique decomposition as multiplication by a nonzero element of the field followed by an automorphism of the field. This correspondence is one-to-one whenever the form is nondegenerate. 1. Introduction. In recent papers by Butts and Pall (3), Butts and Estes (2), and Taussky (4), the representations of a rational multiple cip of binary quadratic form y by another binary quadratic form are studied. In this paper we generalize these results to rational representations of powers of irreducible decomposable forms. We show (Corollary 3.1) that any irreducible decomposable form which does not properly represent zero (for decomposable forms this is equivalent to being nondegenerate) and whose degree is relatively prime to the number of its variables has only a finite number of rational automorphs. Another interesting result (Corollary 1.3) is that a norm preserving linear endomorphism of an alge- braic number field (considered as a linear space over the rationals) which leaves the identity fixed is an automorphism of the field. This result is well known for quadratic fields and (in modified form) quaternion algebras, and provides one of the cornerstones for the theory of compo- sition of binary and quaternary quadratic forms.

In this paper a relationship is established between the concepts of E-continuity and shape for compact metric spaces, thus answering a question raised by Victor Klee. Victor Klee [4] has asked whether there is a relationship between E-continuity and shape. Here we show that, in the category of compact metric spaces, the shape morphisms can be identified in a one-to-one mariner with the limits of certain equivalence classes of c-continuous functions. Throughout this paper all spaces are assumed to be compact metric spaces. If f is a continuous function (map) from one space X to another space Y, [/] will denote the homotopy equivalence class of f. By Cov(Y) we will denote the set of all finite open covers of Y directed by refinement. If A and B are collections of subsets of Y, A 0, if and only if it is a-continuous for some a E Cov(Y) consisting of open balls of radius E). If g is another function from X to Y, f and g are a-homotopic if there is an a-continuous function H from X x I to Y such that, for each x E X, H(x, 0) = f(x) and H(x, 1) = g(x). This defines an equivalence relation on the set of a-continuous functions from X to Y. We say that f and g are a-close if /f(x), g(x) }: x E X I < a. For a E Cov(Y), [X, Y]a denotes the set of a-homotopy classes of a-continuous functions f from X to Y. The equivalence class of such an f is denoted f] a. By lim [X, Y] a we denote the usual representation, in the category of sets and functions, of the limit of the inverse system if X, Y]a P aal a < a E Cov(Y)} where Paa'([f]a') = [f]a for a < a E Cov(Y). Received by the editors June 29, 1973 and, in revised form, November 11, 1973. AMS (MOS) subject classifications (1970). Primary 55D99.

for all x, yER. In this paper we study Lie isomorphisms of a primitive ring R onto a primitive ring R', where we assume that the characteristic of R is different from 2 and 3 and that R contains three nonzero orthogonal idempotents whose sum is the identity. Such isomorphisms will be shown to be of the form o+r, where ois either an isomorphism or the negative of an anti-isomorphism of R into a primitive ring L' containing R' and r is an additive mapping of R into the center of L' which maps commutators into zero. This generalizes a theorem of Hua [1], who obtained the above result in the case that R (= '= L') was the ring of all n X n matrices over a division ring, n> 2. On the other hand, due to our requirement concerning idempotents, we fall far short of providing a general solution to Herstein's conjecture [2] that the result holds for arbitrary simple rings. An important part of our proof consists of a repetition of arguments involving matrix units used by Hua [1], and for the sake of completeness (and also because of the relative inaccessability of Hua's paper) we shall reproduce his proofs in some detail when the occasion demands. The author is especially grateful to Professor Nathan Jacobson for several valuable comments towards improving an earlier version of this paper. In the case of simple rings he indicated how all calculations involving matrix units can be eliminated, and he suggested extending our original result to the more general case of primitive rings.

In my dissertation' I defined a W. set as follows: If n > 1, a W. set is a compact continuum M for which there exists a family F of n elements such that (1) each element of F is an upper semicontinuous collection of mutually exclusive continua which fills up M and is an arc with respect to its elements, and (2) if G is a collection of continua each belonging to some, but no two to the same, collection of the family F, then the continua of the collection G have a point in common and their common part is totally disconnected. In that paper it was stated2 without proof that in the plane there exists a W3 set M whose boundary is the point set B(M) and which has a complementary domain whose boundary, J, contains six limit points of B(M) J but that no W3 set has a complementary domain whose boundary contains seven such points. It is the purpose of this paper to prove this statement. In what follows the space considered will be the plane and if M is a point set the notation B(M) will be used to denote its boundary.

In recent years several papers [1; 3; 4; 5; 6; 7; 9; 10; 11] have appeared dealing with the problem of "Factorisatio numerorum," the number f(n) of representations of an integer n as an ordered product of factors greater than 1. As a result, the basic combinatorial properties of f(n) and the asymptotic behavior of its summatory function are well known. In this paper, I determine the asymptotic behavior of f(n) itself for squarefree n and use the result to determine a "normal" order of f(n) for all n. If n is written as a product of powers of distinct primes pi, then f(n) is evidently a symmetric function of the exponents of the pi's. If n is squarefree, all the exponents are 1 and f(n) can be considered a function of the single variable r, the number of distinct prime factors of n. It is therefore convenient to define, for positive integral r

In the author's paper [1 ] it is shown that if a sequence of orientable n-dimensional generalized closed manifolds (abbreviated n-gcm) converge (n-1)-regularly to an n-dimensional set, then the limit set is also an orientable n-gcm (see Definitions 1 and 2). In this paper a similar result is obtained for manifolds with boundary. Throughout the paper we assume that our sets are imbedded in a compact Hausdorff space S and that the cycles are Cech cycles with coefficients in an arbitrary field which we will omit from our notation for a cycle. All of the basic homology theory needed is in [7] and a knowledge of it will be assumed. We shall use the notation { A } --+A to mean that the sequence of sets A iCS converges to A CS as a limit (see p. 10 of [5]). We shall consider convergence only when all of the sets Ai are closed, and shall not explicitly state this henceforth; the limit set, as is well known, will always be closed. We shall use small latin letters for points, large ones for sets, and script letters for finite collections of open sets which are coverings of S. We shall use "U" for point set union or sum, "n" for intersection, reserving + and - for the group operations. By VAA we shall mean the subcomplex of the nerve of V whose vertices are elements of the covering V that have a nonvacuous intersection with A. If every element of V is contained in some element of V, we shall say that V is a refinement of V, and shall write V >V, or V = V(V); we shall use the notation' IIH. to denote a simplicial projection from the nerve of V into the nerve of V. By U C C V we mean that UC V, and by V>? we mean that the closure of the elements of V form a refinement of V, and by V> *V, we mean that the stars of the elements of V form a refinement ofV. If a cell a of a covering V has a nucleus (the intersection of all the open sets representing the vertices of a) that meets a set B, we say that af is on B; if all the cells with nonzero coefficients in a chain Cr ofZ V are on B, we say that Cr is on B. If the nucleus of a is in the open set Q, we say a is in Q. We shall use

Introduction. Let L'(G) be the group algebra of an arbitrary locally compact abelian group G. It is a regular semisimple self-adjoint commutative Banach algebra with the LI-norm. The space of regular maximal ideals of L'(G) can be identified with the dual group (character group)P=6 of G. The Gelfand transform of a function f in L'(G) coincides then with the Fourier transform] Off. A subalgebra A of L'(G) is called nondetermining if A is not uniformly dense in Co(r) (cf. [2]). It is called irreducible if it does not contain any closed ideal of L'(G) with hull contained in proper closed subgroup of r. In case A does not contain any ideal of L'(G) at all, it is called completely irreducible. In this paper, we will first show that there is a reduction of maximal nondetermining subalgebras to irreducible ones. The reduced algebra will be a closed subalgebra of L'(G') where G' is a quotient group of G. Next we proved that for certain maximal nondetermining subalgebra A, we have S(A) =) c M(A). Finally, we exhibit a maximal subalgebra of L'(Z??) which is nondetermining and which does not belong to the two categories of maximal subalgebras previously known for discrete locally compact abelian groups, namely, a maximal subalgebra of L'(Z*) which is neither associated with an order of G nor gotten from an essential maximal subalgebra of C(E) for a suitable Cantor set E in r. Throughout this paper, we adopt the terminology and notations of Rudin [4] and Hoffman & Singer [3].

Let d be a divisor of a positive integer n. Then d is a unitary divisor if d and nld are relatively prime, and d is a bi-unitary divisor if the greatest common unitary divisor of d and nld is 1. An integer is bi-unitaty perfect if it equals the sum of its proper biunitary divisors. The purpose of this paper is to show that there are only three bi-unitary perfect numbers, namely 6, 60 and 90. A divisor d of an integer n is a unitary divisor if d and nld are relatively prime. A divisor d of an integer n is a bi-unitary divisor if the greatest common unitary divisor of d and nld is 1. Let a(n) be the sum of the divisors of n, let ?*(n) be the sum of the unitary divisors of n, and let ?**(n) be the sum of the bi-unitary divisors of n. We say that N is unitary perfect if a * (N) = 2N. Subbarao and Warren [2] showed that 6, 60, 90 and 87360 are the first four unitary perfect numbers; Wall reported [3] that 146,361,946,186,458,562,560,000 = 2183.547.11.13.19.37.79.109.157.313 is also unitary perfect and later showed [4] it to be the next such number after 87360. Subbarao [1] has conjectured that there are only finitely many unitary perfect numbers. We say that N is bi-unitary perfect if a* * (N) = 2N. The purpose of this paper is to show that the first three unitary perfect numbers, i.e., 6, 60 and 90, are the only bi-unitary perfect numbers. One easily verifies that ov* is multiplicative and that if p is prime and e> 1, then rY**(pe) = ry(pe) = (pe+l l)/(p 1) if e is odd, and or**(pe) = (pe+l l)/(p 1) -pe/2 if e is even. Hence a**(n)?a(n) with equality if and only if every prime which divides n does so an odd number of times. It also follows immediately that a* * (n) is odd if and only if n is 1 or a power of 2; consequently, each odd prime power unitary divisor of n contributes at least one factor 2 to a**(n). Received by the editors June 10, 1971. AMS 1970 subject classifications. Primary lOA20; Secondary lOA99.

In a recent paper [4]2 it is shown that lattices and trees have a common generalization. These systems will be called median semilattices. These semilattices are characterized below by means of segments (??2 and 4) and by means of betweenness (??3 and 4). In ?5 distributive lattices are characterized by means of medians with no assumption made as to the existence of universal bounds (compare Problem 66 of [1]). In another paper [5], median semilattices are characterized in terms of a binary operation and it is shown that these systems can be imbedded in distributive lattices.

In a previous paper [7], we studied the structure of HID spaces. In this paper, we consider the behavior of HID spaces under monotone mappings. The principal result of this paper is that, given an arbitrary compact metric space Y, there is an HID space X and a monotone map f: X-) Y. We also show that an arbitrary HID space can be mapped monotonically onto a space of any preassigned dimension, and that, given an HID space X and a positive integer n, there is an n-dimensional space Y and a monotone map f: Y->X. R. H. Bing showed in [2 ] that each nondegenerate monotone image of a pseudo-arc is a pseudo-arc. The results of this paper show that no similar monotone invariance property holds for spaces of dimension greater than 1. In this paper, all spaces will be compact metric spaces (compacta). We will be dealing with the Hilbert cube, which we regard as being the product of a countably infinite collection of straight line intervals IX = 11 X 12 X 13 X ... , where Ij = [-1/2i, 1/2'].

Introduction. In a recent paper, [l], A. Beurling has shown that the positive translates of an integrable function defined on [0, m0) generate, in a certain sense, at least one exponential of the form e-iz, x > O, Ia < 0, provided that the function does not vanish outside a finite interval. It is the converse problem with which we shall be concerned here; namely, to what extent can the exponentials so generated be used to approximate the given function. We are able to give what amounts to a complete solution. The situation resembles strongly that of Schwartz' theory of mean periodic functions [2]. M. Kahane has shown in [3] (see also [41) how this theory can be presented very simply using the notion of Fourier transform of a mean periodic function. Beurling also made use of this method in the present case; however, we shall find it convenient to exploit this tool more systematically, in closer analogy with Kahane's work. We shall also study our approximations in a topology (the same as the one used in the theory of mean periodic functions) which is simpler than that of Beurling. Beurling based his work mainly on a certain division theorem which states roughly that an entire function is of finite exponential type if it is bounded on a half plane and equal to the ratio of two bounded analytic functions on the complementary half plane. The conclusions we make here follow from a refinement of this given in ?3 which yields an upper bound for the type of such an entire function. It should be remarked that B. Nyman ([7, pp. 28-29]) has established a result similar to the one given here, using, however, a quite different topology. (The referee calls attention to this in his report; although I have since had the opportunity to consult Nyman's work, it was not accessible to me in New York at the first writing of this paper.)

The purpose of this paper is to give a necessary and sufficient condition in order that the S limit set of a point consists of exactly one periodic orbit which is both S-Lyapunov stable and S-asymptotically stable. The general reference for most terms used in this paper is [2]. In this paper we assume (X, T, r) is a transformation group. X will always be assumed to be a uniform space where the Hausdorff topology in X is induced by the uniformity. T will be assumed to be generative, that is, T is isomorphic to Rm X I X C where R is the additive group of real numbers, I is the additive group of integers, m and n are nonnegative integers, and C is a compact abelian group [4]. S will denote a closed replete semigroup in T. The S-limit set of yEX, Sy, is defined bySv = n te,s Cl (ytS) [1 ]. DEFINITION 1. The orbit of a point yEX is said to be uniformly S-Lyapunov stable with respect to BCX provided that for each index a of X there is an index ,3 of X such that if weEBnyt'13 for some t' E T then wt Eyt'ta for all t E S.

In 1936 G. T. Whyburn [8] defined an irreducible mapping to be a mappingf, of a compact metric continuum A onto B, such that no proper sub-continuum of A maps onto B underf. Similarly, assuming A to be only compact and metric, he defined f to be strongly irreducible provided that no closed proper subset of A maps onto B. J. Rozanska [7] independently defined this second type of mapping in 1937. However, neither of these early papers studied these concepts per se, except that Whyburn pointed out that the Brouwer Reduction Theorem does guarantee that either of these two phenomena can be obtained by perhaps reducing A to a subset, A'. In 1939 Whyburn published a paper [9] specifically about these mappings and proved that for f to be strongly irreducible it is necessary and sufficient that the set D, of all points at which f is one-toone, be dense in X. Also proved were certain other characterizations and another theorem which is of special interest here. This theorem fits into a sequence of theorems beginning with a question raised by W. A. Wilson [II] in 1935. As a setting for these theorems, suppose that f is an open mapping of a compact metric space X onto Y. Wilson's question can be asked as follows: if Y is an arc, can f be monotone and irreducible though not a homeomorphism? B. Knaster [5], in 1935, published an example where this is indeed the case. However, in this example the inverses of points of Y are mixed, some being arcs and others indecomposable continua. Knaster then posed the question as to whether there exists an example in which all inverses are homeomorphic, and in particular, are all arcs. The question concerning arcs as inverse sets was answered in the negative by E. E. Moise [6] in 1949 and the more general question is answered in the affirmative (as Eldon Dyer has pointed out) by an example that R. D. Anderson [1] has announced in another connection, in which the inverses of points are pseudo-arcs. In the meantime certain gaps between these two results have been filled in by authors who were concerned with slightly different hypotheses on X, Y, and f. In 1939 Whyburn [9] proved that if Y is unrestricted, f is light

1. In a previous paper [1] a coprime chain was defined to be an increasing sequence { a,, .., ak } of integers greater than 1 which contains exactly one multiple of each prime equal to or less than ak. We let s(n), n> 1, denote the number of coprime chains with largest member n. For convenience we define s(1) = 1. In this paper we will obtain a partial recursion formula for s(n) and an asymptotic formula for log s(n). A table of values of s(n), n < 113, is also provided.

Introduction. The form f=ax2+by2+cz2 +2ryz+2sxz+2txy is properly primitive when (a, b, c, r, s, t) = 1 and (a, b, c) is odd. With f there is associated a determinant d. The greatest common divisor of the cofactors of the elements of d is designated by Q. Then A is defined by d=-2A. For an indefinite form dA must be positive, where Q is chosen so that QA is negative. Also Q' and A' are defined by Q-= 'Q" and A = A'A" where Q" and A" are the greatest powers of 2 dividing Q and A respectively, so that Q' and A' are odd. With f there is associated a reciprocal form F=AX2+B Y2+ CZ2+2R YZ +2SXZ+2TXY, where QA, QB, ***, QT are the cofactors of the elements of d. The purpose of this paper is to show genera containing more than onel class of properly primitive indefinite forms. The method is that which C. L. Siegel used in a specific example communicated to the first of the authors by letter and which is here generalized to include many forms. By employing this method in Lemma 3 of this paper, many genera are explicitly shown containing at least two classes.

The purpose of this paper is to prove that if the primary group G is a direct sum of closed groups, then any two direct decompositions of G have isomorphic refinements. Special cases of this theorem have been proven by Kulikov [10], Kolettis [9], Enochs [1], and Irwin, Richman and Walker [8]. A special case due to Kulikov is now considered classical: If the primary group G is a direct sum of cyclic groups, then any two direct decompositions of G have isomorphic refinements. This result yields rather immediately the following, also due to Kulikov [10]. If G is a closed group, then any two direct decompositions of G admit isomorphic refinements. In [9] Kolettis embedded the two theorems of Kulikov into a slightly more general result. Enochs proved in [8] that any two direct decompositions of a primary group G into closed groups have isomorphic refinements, but it was not known heretofore that a direct summand of a direct sum of closed groups is always itself a direct sum of closed groups. However, it was shown in [1 ] that a direct summand of a direct sum of a countable number of closed groups is necessarily a direct sum of closed groups, and the general isomorphic refinement theorem was established for a group G that can be expressed as a direct sum of a countable number of a closed group. As indicated above, in this paper we shall remove the countability hypothesis. We follow, for the most part, the notation and terminology of [2]. In particular, we use the term "closed" instead of "torsion complete."

F. B. Jones [2] recently gave an example of a Moore space A,, in which there exists a point x such that A,, is not completely regular at x. It is easy to modify the construction used by Jones so as to obtain a Moore space A in which there exist distinct points a and b such that for every real-valued continuous function f on A, f(a) =f(b). Upon applying Urysohn's process of condensation of the singularities of the space A [4], in a manner similar to that used by Hewitt [1], there results a Moore space X on which every real-valued continuous function is constant. Throughout this paper, J denotes the set of positive integers. A sequence is a function on J, and if f is a sequence and n E J, then f, denotes f(n) . By a Moore space is meant a topological space X whose topology has a basis consisting of sets termed regions, satisfying the following condition (axiom 13, that is, parts 1, 2, and 3 of axiom 1, of [3]): There exists a sequence G such that (1) if nE J, G. is a collection of regions covering X, (2) if nCJ, Gnu?1CG., and (3) if r is a region, xer, and yEr, then there exists a positive integer n such that if g CE Gn and x eg, then g C (rx } ) J{ y }. The following characterization of a Moore space will be used in this paper: X is a Moore space if and only if X is a regular Hausdorff space for which there exists a sequence G of open coverings of X such that if U is an open set and

Let Q be a class of topological spaces. For X in Q, let dim X denote the dimension of X defined in terms of finite open coverings. If Q is the class of all normal spaces, then the dimension of spaces X in Q is characterized by the following theorem [2, p. 212]: dimX^n if, and only if, given any closed subset C of X, any continuous mapping of C into the «-sphere Sn has a continuous extension over X with respect to S". The possibility of replacing Sn in the above theorem by a more general space Y was explored in two earlier papers [8; 9]. Such a space Y we call a test space for Q-spaces of dimension n. To be precise, F is a test space for Q-spaces of dimension n if it satisfies the statement: A Q-space X has dimension ^ra if, and only if, given any closed subset C of X, any continuous mapping of C into Y has a continuous extension over X with respect to Y. A space Y is called an absolute neighborhood retract relative to the class Q (abbreviated ANR(C)) if (a) Y is in Q, and (b) whenever Y is imbedded topologically as a closed subset of a Q-space z, then F is a retract of some neighborhood of Y in Z (see [3]). Kodama has shown in [5] that an «-dimensional metric space F is an ANR (metric) if and only if Y is LC". The results of [8 ; 9 ] may be formulated as follows : if Q is the class of all metric spaces, then an «-dimensional, w-LC, metric space Y is a test space for Q-spaces of dimension n if, and only if, Y is an ANR(metric) and is the homotopy type of a wedge of «-spheres (i.e., a union of «-spheres with a single common point). In this paper we shall prove a similar theorem for Q the class of all normal spaces. In [9] we showed that if Y and Z are both ANR(metric) and of the same homotopy type, then F is a test space for metric spaces of dimension « if and only if Z is. Needed in the proof was Borsuk's homotopy extension theorem. Dowker showed in [2, p. 205] that the homotopy extension theorem holds for mappings of a normal space into an ANR (compact metric) provided homotopy is replaced by uniform homotopy. Using this result, we can prove the following lemma

The purpose of this paper is to consider the class of continuous collections of mutually exclusive compact continuous curves in the plane. Throughout this paper we shall denote by G or G with a subscript or superscript a collection of this class with the property that G with respect to its elements as points is a nondegenerate compact closed point set. G has then the significance of being both a collection of continua and a point set itself. By a continuous curve will be meant a nondegenerate locally connected compact continuum. By a continuous collection will be meant a collection which is both upper and lower semi-continuous. By a (simple) chain will be meant a finite collection xi, x2, * *, x. of open discs (i.e. interiors of simple closed curves) such that i,.j exists if and only if j i-ijI ?1 and is the closure of an open disc (i.e. a 2-cell) if it does exist. The xi are called links of the chain. A subchain of a chain c is a chain whose links are links of c. A chain c will be said to simply cover a set M if c* contains M and if for no proper subchain c' of c does the closure of c'* contain M. Two chains will be said to be mutually exclusive if no link of either intersects any link of the other. A collection C' of sets is said to be a (closed) refinement of a collection C of sets if (the closure of) each element of C' is a subset of some element of C. An emanation point of a continuum M is a point which is the common part of each pair of some three nondegenerate subcontinua of M. A hereditary continuous curve is a continuous curve each of whose nondegenerate subcontinua is a continuous curve. It is immediately clear that if G is connected, G contains uncountably many elements and that only countably many can contain triods [1]. Except for a countable number of elements, each element of G must be either an arc or a simple closed curve. We denote the elements of G which are neither arcs nor simple closed curves by g1, g2, g3, * . . From the hypothesis of continuity of G it follows immediately that no element of a connected G contains a 2-cell.

A considerable amount of work has been done on n-person games in which all coalitions with less than n-1 players are totally defeated. Maschler [3] has studied the structure of their bargaining sets, while Lucas, in two separate papers [1], [2] has given a symmetric solution to such games and studied the behavior of solutions for such games in partition function form. In this paper, we propose to give all discriminatory solutions to such games. We consider here n-person games, v, satisfying

In a recent paper the first author gave an explicit construction of a sequence of polynomials having their zeros on the unit circumference which converge boundedly to a given bounded zero-free analytic function in the unit disk. In this paper we find the best possible uniform bound for such approximating polynomials and construct a sequence for which this bound is attained. The method is also applied to approximation of an analytic function in the unit disk by rational functions whose poles lie on the unit circumference. Some open problems are discussed. Let C denote the unit circumference in the z-plane and D its interior. The term C-polynomial shall mean a polynomial all of whose zeros lie on C. In [6] the first author unified the results of MacLane [5] and Chui [2 ] by proving THEOREM A. Let f(z) = 1 +c+c2z 2+ * * * be a zero-free holomorphic function in D. Then there exists a sequence of C-polynomials Pn(z) assuming the value one at z =0 which converges to f(z) uniformly on every closed subset of D. If in addition M-supizi

A stronger form of semi-aposyndesis is defined and used to characterize those compact, metric continua which are hereditarily locally connected. In [1], Bennett raised the question as to whether strongly aposyndetic continua are necessarily hereditarily locally connected. The purpose of this paper is to give an affirmative answer to the question and to prove the somewhat stronger theorem of the title. A continuum M is strongly aposyndetic if M is decomposable and whenever H and K are proper subcontinua such that M=HUK, then H and K are aposyndetic. The continuum M is strongly semi-aposyndetic if M is aposyndetic and whenever H and K are proper subcontinua such that M= HUK, then H and K are semi-aposyndetic. Since a strongly aposyndetic continuum M is easily seen to be aposyndetic it is not necessary to insist that M be aposyndetic in the definition. However, in the semi-aposyndetic case, requiring M to be decomposable does not insure that M is aposyndetic which it must be in order for the theorems of this paper to be valid. The method of proof will be to first show that an aposyndetic continuum in which every subcontinuum with nonvoid interior is semi-aposyndetic is necessarily hereditarily locally connected. Secondly, it will be proved that a strongly semi-aposyndetic continuum has the property that every subcontinuum with nonvoid interior is semi-aposyndetic. These two results together will complete the proof. Throughout, a continuum is to be a compact, metric, connected topological space. A continuum M is aposyndetic at p where p is a point of M if for every point q in M-p there exists an open set U in M and a subcontinuum H in M such that p E Uc Hc: M-q. If M is aposyndetic at p for each p in M then we say that M is aposyndetic. A continuum M is semi-aposyndetic at {p, q} where p and q are distinct points of M if there is Received by the editors July 15, 1971. AMS 1970 subject classifications. Primary 54F20, 54F50.

In this paper we generalize the notion of exact functor to an arbitrary category and show that every small category has a full embedding into a category of all set-valued functors on some small category. The notion of exact is such that this result generalizes the author's exact embedding of regular categories and, indeed, Mitchell's embedding of abelian categories. An example is given of the type of diagram-chasing argument that can be given with this embedding. Introduction. In [Barr] we proved that every small regular category has an exact embedding into a set-valued functor category (C, 5) where C is some small category. In this paper we show that the same result can be proved for any small category when the definition of exact functor is slightly extended. This new definition will agree with the previous one when the category is regular. The embedding which results can be used to chase diagrams in completely arbitrary categories, much as Mitchell's theorem does for abelian categories. As an illustration of this, we derive a theorem of Grothendieck on the descent of pullbacks. 1. Universal regular epimorphisms. In any category a map is called a regular epimorphism if it is the coequalizer of some parallel pair of maps. One usually defines /: X—>Y to be a universal regular epimorphism if for every Y'-+ Y, the fibred product Y' x r X exists and the projection Y' xr X-* Y' is always a regular epimorphism. It is gradually becoming clearer that the universal regular epimorphisms are the "good" epimorphisms. In fact a good case could be made that these should be termed the quotient mappings. In a regular category, every regular epimorphism is universally so. In [Verdier] it is shown how every category X has a full embedding into a topos E where E is the category of sheaves for the so-called canonical topology on X. A brief description of E and the embedding R follows. Received by the editors May 12, 1972. AMS (A/OS) subject classifications (1970). Primary 18A25, 18B15.

Let F n {F_n} be a free group of rank n n . In a recent paper R. Horowitz has shown that for n ≦ 3 n \leqq 3 the ideal I n {I_n} in the ring of special linear characters of F n {F_n} consisting of those polynomials in the characters which vanish for all representations of F n {F_n} by subgroups of S L ( 2 , C ) SL(2,C) is principal. In this paper the case n = 4 n = 4 is investigated; it is shown that for n > 3 , I n n > 3,{I_n} is not principal.

The Cauchy condition for convergence of a contractive iteration in a complete metric space is replaced by an equivalent functional condition. Many generalizations of the Banach-Neumann contractive mapping principle follow. Introduction. Let X be a complete metric space and let f: X-*X be a contractive mapping or contraction, i.e., d(f (x), f (y)) O. Criteria for the sequence of iterates {xj to be Cauchy are then of interest, for if it is Cauchy one can easily prove it converges to a unique fixed point x"' of the mapf on X. Many papers have presented such criteria, especially since the important paper of Rakotch [1]. In this note we first give in ?1 a necessary and sufficient condition that a sequence of iterates be convergent. This condition is used in ?2 to provide a criterion for the convergence of the iteration from an arbitrary initial point in X. The condition is then restated in a more familiar functional form, and some corollaries are then drawn in ?3. 1. Main theorems. The following results, while stated for complete metric spaces, are readily extended to separated complete uniform spaces, as in [2, p. 197]. We shall use the following notation. For any pair of sequences xt, and y,, in X with xn#y", we write dn = d(xn, yr) and A,, = d(f (xn),f (y0))/d. We then have the following theorem. THEOREM 1.1. Let X be a complete metric space. Let f: X--X with d(f(x),f(y)) O. Then x,,l-)+X,, in X, with xc, a unique fixed point off, ifjfor any Received by the editors January 2, 1973. AMS (MOS) slubject classifications (1970). Primary 54H25; Secondary 47H 10.

The "generating hypothesis" of Freyd implies vari- ous fascinating statements in stable homotopy, particularly about the stable homotopy ring of spheres and the modules over this ring. The present paper extends some of these consequences. It also investigates the question of whether a possible counterexample to the hypothesis might be detected by its action on homology groups ; for this, we find a connection with work of Joel Cohen on coherent graded rings. In his basic paper (2), P. Freyd has put forth a conjecture, known as the "generating hypothesis", which says that if a map/: X—> Y, between finite, connected complexes X and Y, induces the zero map on stable homotopy groups, then/is stably null-homotopic. The conjecture, unresolved at the time of this writing, is remarkable, because of the wealth of statements in stable homotopy theory which are known to follow from it. Some of these statements have been verified in many cases (see, for example (4)), but many of them, while of great interest, remain a complete mystery. They may be equivalent to the conjecture itself; in fact, various equivalent statements are now known (5). One of these statements, which is a consequence of the "generating hypothesis", asserts that if a and s are elements of/?-torsion in the stable homotopy ring of spheres, there is a nonzero y which is simultaneously divisible by a and s (see Proposition 9.3 in (2)). The purpose of the present paper is to study extensions and generalizations of this statement and to prove related theorems which follow without invoking the presently un- proved "generating hypothesis". We first prove a mild generalization of this particular statement for the stable homotopy ring G^=Trf(S°), and then investigate conditions which assure a similar statement for the stable homotopy of a finite complex X. Next, we prove that either there are un- countably many counterexamples to one of the consequences of the "generating hypothesis", or at most finitely-many elements are unrelated to other independent elements via the existence of nonzero common

In this paper we construct an upper semicontinuous decomposition of En (n_3) into points and tame arcs such that the associated decomposition space is distinct from En. The purpose of this paper is to construct an upper semicontinuous decomposition G of En, n>3, into points and tame arcs such that the decomposition space EnIG is topologically distinct from En. For n=3, the example is a modification of R. H. Bing's dog bone space [2]. Unlike the dog bone space, it is not difficult to distinguish our decomposition spaces from Euclidean space. The main idea for the construction of these decomposition spaces was communicated to us by R. D. Anderson during his visit to the University of Texas in March, 1972. Anderson's idea was roughly the following: Take two disjoint wild Cantor sets in En. To each point in one Cantor set correspond a unique point in the other Cantor set and join the two points with an arc that is locally polygonal modulo its end points. The collection of arcs thus obtained can be constructed so that its union is homeomorphic to the product of a Cantor set and an arc. Such arcs for n>4 are tame and, hopefully, if the Cantor sets are wild enough and the pairings are chosen cleverly, the resulting upper semicontinuous decomposition of En will have a decomposition space distinct from En. The difficulty of proving that such decomposition spaces are not En lies in finding a topological property of En not shared by the decomposition spaces. We use the following elementary property of En. THEOREM 1. If C is a Cantor set in En (n _ 3), U is an open set containing C, andf and g are maps from a 2-cell D into En, then there exist maps f' and g' from D into En such that f'lf-1(EnU)=f If`1(EnU), g'lg-1(En U) =gjg`(EnU), fI(f(U)) c U, g'(g-(U)) c U and f'(D ng'(D) nx C_ = Received by the editors July 24, 1972. AMS (MOS) subject class/ifcations (1970). Primary 57A15, 57A10, 54A30; Secondary 57A35, 57A45.

In a recent paper (Square roots in locally euclidean groups, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 446-449), A. M. Gleason proved that, in a locally euclidean group G which has no small subgroups, there exist neighborhoods M and N of the neutral element e such that every element in M has a unique square root in N. The author clearly considered this result to be a step towards proving that, in such a group, some neighborhood of e is entirely filled by oneparameter subgroups of G. We shall establish here that he was justified in his expectation. As a matter of fact, our present proof requires only the local uniqueness of the square root, not its existence. This local uniqueness is established in the second part of the proof of Theorem 4 in Gleason's paper; and it may be observed that this part of the argument is independent of the assumption that the group be locally euclidean: it works equally well under the weaker assumption that the group be locally compact. We therefore have the following result

In this paper a stronger form of aposyndesis is defined and continua with this property (strongly aposyndetic) are shown to be countable-set aposyndetic. Although every continuum which is a product of nondegenerate continua is countable-set aposyndetic, it is established that no product of nondegenerate continua is strongly aposyndetic. Throughout this paper a continuum is a compact connected metric space. Let M be a continuum. If N is a subcontinuum of M, the interior of N in M will be denoted by int N and the boundary of N in M by Bd N.2 A continuum M is aposyndetic at a point p if for each point q in M{p} there is a subcontinuum N in the complement of {q} such that peint N [3]. If the {q} above is replaced by a finite (countable closed) set then M is said to be finitely (countable-set) aposyndetic at p. A continuum is aposyndetic (finitely aposyndetic) (countable-set aposyndetic) if it has the given property at each point [7]. This familiar property can be strengthened as follows. DEFINITION. A decomposable continuum M is strongly aposyndetic provided that for any pair of proper subcontinua H and K such that M=HUK, each of H and K is aposyndetic. It is easily seen that each strongly aposyndetic continuum is aposyndetic and there are numerous elementary examples of aposyndetic continua which fail to be strongly aposyndetic. The following two theorems established sufficient conditions for subcontinua of a strongly aposyndetic continuum to be aposyndetic. The first theorem is an immediate consequence of the definition. THEOREM 1. If A is a nonseparating subcontinuum of a strongly aposyndetic continuum M and int A $ 0, then A is aposyndetic. Received by the editors October 15, 1970. AMS 1970 subject classifications. Primary 54F20, 54F15; Secondary 54B10.

Swingle [7]1 has given the following definitions. (1) A continuum M is said to be the finished sum of the continua of a collection G if G = M and no continuum of G is a subset of the sum of the others.2 (2) If n is a positive integer, the continuum M is said to be indecomposable under index n if M is the finished sum of n continua and is not the finished sum of n +1 continua. Swingle has shown [7, Theorem 2] that if n is a positive integer and the continuum M is indecomposable under index n, then M is the finished sum of n indecomposable continua. The author has shown [2, Theorem 1] that if n = 2 and the continuum M is indecomposable under index n, and G is a collection of n indecomposable continua whose finished sum is M, then G is the only such collection. In the present paper, it is shown that for a compact continuum, this theorem holds for any positive integer n. Also, there is given a necessary and sufficient condition that a compact continuum be indecomposable under 'index n. An indecomposable continuum can be described as a nondegenerate continuum which is indecomposable under index 1. If n =1, then in order that a continuum M be indecomposable under index n, it is necessary and sufficient that M contain n+2 points such that M is irreducible about any n+1 of them.3 Swingle [7] has shown that it is impossible, in a certain manner, to generalize this theorem. Theorem 3 of the present paper might be considered a generalization of the necessary condition of the above theorem. However, it is easily seen that the converse of Theorem 3 is not true. Theorems 1-5 are proved on the basis of R. L. Moore's Axioms 0 and 18. Hence these theorems hold in any metric space.4

Subsequently Leighton [3] proved that conditions (2) were sufficient without the restriction q be positive for sufficiently large x. In this paper we prove analogous theorems for the linear equation of order 2n (ry(n)) (n) + ( -1)n-lqy = 0. There will be no sign restrictions on the function q. For n= 1, the earlier results of Wintner and Leighton will be contained in our theorems. The results of this paper parallel earlier results of the author for a fourth-order equation with middle term [1]. Throughout r and q denote continuous, real-valued functions on a ray [a, oo) and r is assumed to be positive-valued. If the real-valued function y has n continuous derivatives and ry(n) has n continuous derivatives, then we define L(y) by

In this paper, we shall make use of LEMMA 1.1. (1.1) is a determinate Hamburger moment sequence in case En=O 1/J an= + oo [2; 3]. Presented to the Society, April 22, 1955; received by the editors September 1, 1955 and, in revised form, November 17, 1955. 1 The material in this paper is contained in a thesis submitted to Northwestern University and prepared under the direction of Professor Walter T. Scott. 2 Numbers in brackets refer to references at the end of the paper. 1040

It would be of some interest to know exactly which primes divide the numerator or denominator of o-2n(v), if v is rational and oa2n(v) is reduced to its lowest terms. Restricting ourselves to the prime 2, and using induction on (1.1), we shall find in this paper the exact power of 2 dividing o-2n(v) when v is a rational number a/b, a odd and b even. This will extend the results of [2 ] where the same problem was solved for v one half an odd integer. In this paper we shall also prove some congruences (mod 2), (mod 4), and (mod 8). We note that this type of problem has been considered for other sequences of rational numbers. For the well-known Bernoulli numbers B2n, for example, it has been proved [1] that

In this paper it is shown that a commutative semisimple ring is isomorphic to a direct product of fields if and only if it is hyperatomic and orthogonally complete. In this paper we give a necessary and sufficient condition for a commutative semisimple ring R (i.e., R has no nonzero nilpotent element) to be isomorphic to a direct product of fields. In particular, we show that hyperatomicity and orthogonal completeness is such a necessary and sufficient condition. It is well known that without these conditions R is isomorphic to a subring of a direct product of fields [1, p. 16]. We would like to emphasize that, in what follows, R stands for a commutative ring with no nonzero nilpotent element. Thus, in particular, for every element x of R, (1) X2 =O if and only if x = O. We first prove several lemmas. Lemma 1 below, generalizes the corresponding result for Boolean Rings [2, p. 154]. LEMMA 1. The ring R is partially ordered by : where for every element x and y of R, (2) x 5 y if and only if xy = x2. PROOF. Since xx=x2 it follows from (2) that x gx. Thus, ? is reflexive. Moreover, if x9y and y

We establish under very general conditions that a space may be changed by adding one cell, so that the stable homotopy module becomes infinitely generated over the stable homotopy ring. We give an application to the study of ideals in the stable homotopy ring. The stable homotopy groups of a finite CW-complex form a graded module over the stable homotopy ring G*. There has been some interest in these modules, largely due to the "Generating Hypothesis" of [2], which asserts that the functor which associates, with each finite complex, its stable homotopy module, is an imbedding in the category-theoretic sense. Although we do not know if this hypothesis is true, we now know that it has the following statements as corollaries. (1) If X is a finite complex, the stable homotopy module of X, irs(X), is finitely-generated over G*, when and only when X has the stable homotopy-type of a bouquet of spheres. (2) G* is totally noncoherent in the sense that no nontrivial, finitely-generated, ideal of the ring G* is finitely presented. In spite of our ignorance about the generating hypothesis, I established in earlier papers, [3] and [4], that ig(X) is often infinitely-generated. In fact, it is shown in [4], that if a finite complex X has a nonzero rational homology group in some positive dimension, then we may add one cell to some suspension of X, and thus obtain a complex with infinitely-generated stable homotopy module. The purpose of the present paper is to prove the following stronger theorem. THEOREM 1. Let X be a finite complex. If there is an odd prime p such that H.(X, Z,) $0 for some n>O, then we may find i>O and a: Sn-l --+;iX so that (2iX) kJ,eem has infinitely-generated stable homotopy module. In a sense, excluding 2-torsion, statement (1) above is within one cell of being true. Now, one may easily see that statement (2) actually follows from statement (1). We shall thus also obtain Received by the editors October 2, 1969. AMS 1969 subject classifications. Primary 5540; Secondary 5530.

In the second paper in this series [2; 3], it was demonstrated that under suitable topological conditions on a space S (which are in particular always satisfied for a manifold), the problem of reconstructing S from its group G of homeomorphism onto itself may be reduced to the problem of reconstructing the points of S from G. If S can be reconstructed then every automorphism of G is inner, and it was shown in this way that every automorphism of G is in fact inner when S is the unit interval; this is the theorem of Fine and Schweigert [I]. The present paper is devoted principally to proving the FineSchweigert theorem for the disc. Given a space S topologically equivalent to the interior of the unit circle in the plane, the point set of S will be canonically reconstructed from the group G of homeomorphisms of S onto itself. A principal tool is the classical theorem that if g is an orientation-preserving homeomorphism of finite order of the disc S onto itself, then S may be mapped homeomorphically on the interior of the unit circle in such a way that g becomes a rotation, [4]. Since a homeomorphism of odd order is necessarily orientation-preserving, it follows that an element of odd order (different from one) of the group G of homeomorphisms of S onto itself has precisely one fixed point, which will be called its center. If x is any point of S, let x' denote the set of all elements of G of finite odd order (different from one) having x for center. The collection S' of all sets x' is in canonical one-one correspondence with the point set of S. Our problem will be at an end if S' can be exhibited as a structure derived from the purely algebraic structure G. To this end it is sufficient to be able to decide within the structure of G when two elements of odd order (different from one) have distinct centers. This occupies the remainder of the paper. Henceforth, when an element of G is said to be of odd order, it will be understood that the order is different from one.

where the 'indicates that the index k runs through integers relatively prime to n. The degree of Fn(x) is +(n), Euler's totient. This paper determines the resultant p(Fm, Fn) of any two cyclotomic polynomials Fm and Fn. Explicit formulas are given which show that if m$n the resultant is either 1, -2, or a prime power. For the case m > n > 1 the results agree with a formula derived by Diederichsen [3, Hilfssatz 2] in a paper on group representations (see Theorem 4 below). Our proof is different from and somewhat simpler than that of Diederichsen; it is based on the following lemma on decompositions of reduced residue systems which the author has recently used to relate Gauss sums and primitive characters [1, Lemma 6].

A necessary and sufficient condition on the endomorphism ring of a module for the module to have the finite exchange property is given. This condition is shown to be strictly weaker than a sufficient condition given by Warfield. The class of rings having these properties is equationally definable and is a natural generalization of the class of regular rings. Finally, it is observed that in the commutative case the category of such rings is equivalent with the category of ringed spaces (X, M) with X a Boolean space and M a sheaf of commutative (not necessarily Noetherian) local rings. In all that follows we will consider modules and morphisms as being over a ring S. Given two direct decompositions K=MeX= ?3 _ {A i E I} of a module, we say that they can be exchanged at M if there are submodules Ac Ai (i e I) such that K=ME() {Aji eI}). A module M is said to have the n-exchange property if any pair of decompositions K=M'eX= ? :,{A i E I} with M'_M and card(I)_n can be exchanged at M'. The module M has the (finite) exchange property if it has the n-exchange property for all (finite) cardinals n. In a fundamental paper [1], Crawley and Jonsson define the exchange properties and use them to prove theorems on isomorphic refinements of direct decompositions of modules. Warfield has shown [3] that an indecomposable module has the exchange property if and only if its endomorphism ring is local. In an attempt to generalize this theorem he has shown recently [4] that if R is Ends(M) and J is its Jacobson radical, then M has the finite exchange property if every principal left ideal of R/J is of the form Re+J where e is an idempotent element in R. In the present paper we will give another condition on R which is, in fact, necessary and sufficient for M to have the finite exchange property and furthermore is strictly weaker than Warfield's. We will frequently use the fact that a direct decomposition K=AEB of a module is naturally related to an idempotent endomorphism 7r of K which leaves the elements of A fixed and has B as its kernel. This will be Presented to the Society, June 28, 1971; received by the editors January 24, 1972. AMS 1970 subject classqicfations. Primary 16A30, 16A32.

In this paper an example is given of a ring with left global dimension 2 having the property that the singular submodule of any R-module A is a direct summand of A. Although the example given is quite specific, the methods can be used to construct a fairly large class of these rings. In this paper, all rings are assumed to be associative with an identity element, and all modules will be unitary left modules. An R-module A is said to split if the singular submodule, Z(RA), is a direct summand of A. R is called a splitting ring if every R-module splits (see [1], [3], and [7]). In [3] Cateforis and Sandomierski have shown that every commutative splitting ring has left global dimension _1. M. L. Teply [7] has shown that if the commutative hypothesis is dropped, then every splitting ring must have left global dimension

Introduction. The object of this paper is to characterize the clans (compact connected Hausdorff topological semigroups with an identity element) which are homeomorphic to a unit interval and which have a nondegenerate kernel (minimal two-sided ideal). The corresponding case when the kernel is degenerate has been characterized in a paper by H. Cohen and L. I. Wade [2] together with an earlier paper by Mostert and Shields [5]. In a topological semigroup T, K(T) or K denotes the kernel of T. The symbol u is reserved to denote an identity element. The term "standard thread" will mean a clan with zero which is homeomorphic to a unit interval and whose endpoints are its zero and identity element. In a standard thread T with identity element u and zero 0, for a, bCT, [a, b] will denote the interval from a to b, (or b to a) inclusive and a ? b will mean aC [0, b], with a x implies aRx (closed). We denote by Ra the set {x aRx }. The analog of Theorem I where R satisfies (2)' (a, b, cCX, aRb implies caRcb (left congruence) instead of (2) is also true. I would like to express my appreciation to Professor H. Cohen and Professor R. J. Koch for their assistance in the preparation of this paper.

In this paper a theory of pseudo-complements is developed for posets (partially ordered sets). The concepts of ideal and semi-ideal are introduced for posets and a few results about them are obtained. These results together with known results about pseudo-complements in distributive lattices lead to the main results. It is proved that if in a pseudo-complemented semilattice or dual semilattice every element is normal, then it is a Boolean algebra. Using this result new proofs for two known theorems are obtained. The existence of maximal ideals in posets is established and it is shown that the dual ideal of dense elements of a poset with 0 is the product of all the maximal dual ideals. Already, there exists a theory of pseudo-complements for lattices. Frink [5 ] has obtained a generalisation of the theory for semilattices. In this paper we extend some of the results of Frink [5] and Balachandran [1] to posets (partially ordered sets). We obtain these extensions by using the concept of semi-ideal, which we define in ?2. This paper consists of three sections. In ?1 we summarise some known results which we use in later sections. ?2 deals with some of the properties of semi-ideals and ideals in posets. Our definition of poset ideal is different from that introduced by Frink [4]; however in a lattice our definition is equivalent to the usual definition. Using the results obtained in ?2, we develop a theory of pseudo-complements for posets in ?3. 1. Preliminaries. We shall denote the ordering relation in a poset by ieI ai and rliET ai respectively. When A is finite, say, A = { a,, a2, * , an }, the lattice-sum and the lattice-product of the as are denoted by al+a2+ * . . +a. and a,*. a. respectively. The Received by the editors February 3, 1970. AMS 1969 subject classifications. Primary 0620; Secondary 0630, 0635.