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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.

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

In a recent paperf entitled A note on the linear fractional group, Todd obtained an abstract definition for the group LF(2, 2) in terms of n+2 generators. Apparently he gave no consideration to the question of the independence of the defining relations, for they can be considerably simplified. First, in view of the condition RSi = Si+iR (which is the same as i£S0^~* = Si), the three generators U, R and S0 are sufficient to generate the entire group. If we give a definition in teims of these three generators alone, the relations

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].

In [2], Tosiro Tsuzzuku gave a proof of the following:THEOREM. Let G be a doubly transitive permutation group of degree n, let K be any commutative ring with unit element and let p be the natural representation of G by n × n permutation matrices with elements 0, 1 in K. Then ρ is decomposable as a matrix representation over K if and only ifn is an invertible element of K.For G the symmetric group this result follows from Theorems (2.1) and (4.12) of [1]. The proof given by Tsuzuku is unsatisfactory, although it is perfectly valid when K is a field. The purpose of this note is to give a correct proof of the general case.

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.

IN this paper Haldane points out (p. 142) a difference between his results for the mean and variance of x2 in a 2 x n-fold contingency table when the expectation p is fixed and the results obtained by me in my paper (Annals of Eugenics, Vol. VII, part III, p. 211). The difference is that I have (n 1) throughout where Haldane has n. Haldane writes, "my own results would appear to be slightly more accurate than Cochran's", which might, I think, give the impression that both Haldane's results and mine are only approximations. In fact, both results are mathematically exact, the difference between them being one of definition of x2. My paper is almost entirely concerned with the distribution of x2 when the expectation p is not known. In the results which I gave for the distribution of x2 when p is known, I retained the term S (x -x-)2 in the numerator of x2 instead of S (x np)2, to facilitate comparison between this and my other results. Thus my x2 has (n 1) degrees of freedom, whereas Haldane's x2 has n degrees of freedom. Unfortunately I did not emphasize this point in the passage concerned, and as it may have appeared misleading to others besides Haldane, I welcome this opportunity of drawing attention to it. Haldane's x2 is, of course, the one which is normally appropriate in testing the departure from independence when the expectation is known.

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.

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.

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

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.

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

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.

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).

Papers that have been written on the use of the rank order statistic for the two-sample problem under the Lehmann alternatives have focused on the use of rank tests to determine whether there is a difference in the two populations. In particular, the test proposed by Savage (1956) is the rank test that is usually suggested to be used in this situation. In this paper a Bayesian analysis of the problem based on the rank order statistic is presented. Suppose that a random sample of m observations is obtained from distribution 1 and an independent random sample of n observations is obtained from distribution 2, and N = m+n. The rank order statistic will be denoted by z = (z1, ...,ZN), where zi is 0 if the ith smallest observation is from distribution 1 and zi is 1 otherwise (i -1, ..., N). Let F(x) and G(x) be the distribution functions of distributions 1 and 2, respectively, both assumed to be continuous. If G(x) is the Lehmann alternative 1{1 F(x)}k(k > 0), which holds, for example, if the population distributions are both exponential or both Weibull, then the rank order probability is

In a previous paper, we developed lower and upper bounds for the generalized hypergeometric functions p F q , p = q , p = q + 1, and certain confluent forms under appropriate restrictions on the variable and parameters. In the present paper, we extend these notions and obtain similar inequalities for certain generalized hypergeometric functions of two variables.

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.

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.

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.

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].

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

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 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

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.

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)).

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'].

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

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.)

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.

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 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.