Norman L. Alling

On the existence of real-closed fields that are _{}-sets of power ℵ_{}

Introduction. In the theory of 7la-sets three main theorems stand out: I. An 77-set is universal for totally ordered sets of power not exceeding Ka. II. Two 71a-sets of power R. are isomorphic. III. If R. is regular and if Ea 0 then these three theorems hold for the category of totally ordered Abelian groups and order preserving (group) isomorphisms; the special group being totally ordered, Abelian, divisible, and an 77a-set. Erdos, Gillman, and Henriksen [8] (see also Gillman and Jerison [11]) proved that if a> 0 then I and II hold for the category of totally ordered fields and order preserving (ring) isomorphisms; the special field being real-closed and an 7a-set. It was also shown in [8] that III holds for this category and special object if a = 1. However in case a > 1, III was left open both in [8] and in [11]. The initial aim of these researches was to show that, assuming a> 0, K, regular, and Ea 0. Let R { G } denote the field of formal power series with exponents in G and coefficients in R, the reals. R{G } is an 7la-set but its power exceeds K. Let R { G } a = {f ER { G }: the support of f is of power less than a }. Then R{G } a, again a real-closed field, is an 77a-set, and is of power K,. The only difficult point in these verifications was the proof that R { G } and R{G}a are 77a-sets. The proof arrived at by the author did not involve the multiplication in these fields, but depended wholly on their structure as a

On exponentially closed fields

It is well known [4] that the non-Archimedean residue class fields K of the ring of continuous real valued functions on a space are realclosed and -jl-sets. It does not appear to be known that the exponential function in the reals induces an exponential function in K (definitions to follow); thus K is exponentially closed. The property of being exponentially closed is a new invariant which will be applied to totally ordered fields in this paper. A totally ordered field K will be called exponentially closed if (i) there exists an order preserving isomorphism f of the additive group of K onto K+, the multiplicative group of positive elements of K, and (ii) there exists a positive integer n such that 1 +1/n <f(1) <n; such an isomorphism will be called an exponential function in K. In ?0 Archimedean exponentially closed fields will be considered, the rest of the paper being devoted to the non-Archimedean case. In ?1 some necessary conditions for a non-Archimedean field to be exponentially closed will be given, followed in ?2 by some examples. In ?3 a set of sufficient conditions will be given, followed by an example. A totally ordered field K will be called root-closed if K+ is divisible. Clearly exponentially closed fields and real-closed fields are rootclosed.

CONWAY'S FIELD OF SURREAL NUMBERS

Conway introduced the Field No of numbers, which Knuth has called the surreal numbers. No is a proper class and a real-closed field, with a very high level of density, which can be described by extending Hausdorff s ri( condition. In this paper the author applies a century of research on ordered sets, groups, and fields to the study of No. In the process, a tower of sub fields, £No, is defined, each of which is a real-closed subfield of No that is an T/£-set. These fields all have Conway partitions. This structure allows the author to prove that every pseudo-convergent sequence in No has a unique limit in No. 0.0. In the zeroth part of J. H. Conways book, On numbers and games (6), a proper class of numbers, No, is defined and investigated. D. E. Knuth wrote an elementary didactic novella, Surreal numbers (15), on this subject. Combining the notation of the first author with the terminology of the second, we will call No the Field of surreal numbers. Following Conway (6, p. 4), a proper class that is a field, group,... will be called a Field, Group,_We investigate this Field using some of the methods developed in the study of ordered sets, groups, and fields over the last 100 years or so. (A short, partial bibliography on this subject will be found at the end of the paper.)1 0.1. Let Tbe a partially-ordered class. It will be called totally-ordered (= linearly- ordered = simply-ordered) if for all x and y in T then x 0 and y ^ 0 in T imply xy > 0. T will be called Dedekind complete if, given any bounded subset B of T, it has a l.u.b. in T. It is well known that, up to isomorphism, , . the only Dedekind complete totally-ordered field is the field R of all real numbers.

Rings of continuous integer-valued functions and nonstandard arithmetic

0. Introduction. In this paper rings of continuous integer-valued functions are studied, with particular attention paid to their maximal residue class domains. These domains correspond bijectively to minimal prime ideals, rendering the space of these ideals of particular interest. Since these domains are either the integers or are nonstandard models of the integers, questions about nonstandard arithmetic will also be considered. In ?1 the space of minimal prime ideals of C(X, Z), the ring of continuous functions from a nonempty Hausdorff space X into Z, the ring of integers, is showed to be homeomorphic to 6X (1.2), the Boolean space of the algebra of open-and-closed sets of X. The maximal ideal space of C(X, Z) is shown to map continuously onto 6X (1.3). The space, 6OX, of points of 3X that give rise to integer residue class domains, is studied in ?2. The map of X into boX strongly resembles the realcompactification injection [GJ]. A representation theorem of C(X, Z) over 3OX is also given (2.4). It is shown in ?3 that points in 3X 6OX give rise to Z, a nonstandard model of Z (3.1). Here some of the relevant background material in model theory is discussed. The algebraic theory of nonstandard arithmetic is studied in ?4. In ?5 we return to study Z, its maximal ideal space, and its quotient field Q, which is a nonstandard model of the rational field Q. In ?6, the most technical section of the paper, the valuations of Q associated with maximal ideals of Z are computed (6.3). The value groups that arise are analysed ((6.4), (6.5), and (6.6)), followed by some rather striking results in case the maximal ideal in question is principal. The ideals of Z are analyzed in ?7 along classical lines: i.e., we proceed from the study of maximal and prime ideals, through the study of primary ideals, to a decomposition theorem for ideals in terms of primary ideals (7.4). Ideals in C(X, Z) are decomposed in ?8, first into coprimary ideals (8.4), and then into primary ideals (8.9). In the process, the sets of maximal, prime, coprimary, and primary ideals of C(X, Z) are analyzed. In ?9 some model-theoretic results are obtained on the residue class fields of C(X, Z), the principal result being that any such field is elementarily equivalent

Global ideal theory of meromorphic function fields

It is shown that the ideal theories of the fields of all meromorphic functions on any two noncompact Riemann surfaces are isomorphic. Further, various new representation and factorization theorems are proved. Introduction. Throughout this paper let X and Y denote noncompact (connected) Riemann surfaces. Let A(X) (or A for short), denote the ring of all analytic functions on X, and let F(X) (or F for short), denote the field of all meromorphic functions on X. In 1940 Helmer [10] studied divisibility properties in A(C), laid the foundations for its ideal theory, and proved that every finitely generated ideal in it is principal. (See [2, pp. 24-28] for a brief history of the subject from 1940 to 1966.) In 1952-53 Henriksen [11], [12] investigated the maximal and prime ideals of A(C), finding-among other things-that each prime ideal is contained in a unique maximal ideal. An ideal of a ring will be called local if it is contained in a unique maximal ideal; thus Henriksen proved that each prime ideal of A(C) is local. In 1948 Florack [7] proved essentially that X is a Stein manifold. Using her theorem, the investigation of the ideal theory of A(X), for X c C, was gradually generalized to arbitrary X. In 1963 the author [1] showed that if M is a maximal ideal of A then the ring of quotients, AM, is a valuation ring. At that time the value group of AM was also investigated. Using classical methods of commutative algebra, one can make a very complete analysis of the local ideals of A. The initial aim of this research was to learn more about the decomposition of an ideal I of A as an intersection of local ideals. In trying to extend local knowledge to obtain global results it became evident that some topology on the set specm A of maximal ideals was needed. The author turned, naturally, to the Zariski topology on specm A. X is, in a natural way, identifiable with a subset of specm A. Let X0 be the topology induced on X by this identification; it will be called the zero set topology on X. It is obvious that X0 is a much coarser topology than X. The author was surprised to learn (1.3) that X0 and Y0 are always homeomorphic. One possible inference to be drawn is Received by the editors December 1, 1978. AMS (MOS) subject classifications (1970). Primary 13A15, 30A98.

An application of valuation theory to rings of continuous real and complex-valued functions

the residue class fields of these rings are always totally ordered and, over noncompact spaces, may be non-Archimedean. Kohls [22] showed that the residue class domains of these rings are also totally ordered. In some recent researches [1 ; 2] the author has exploited a fact, known to Baer [3] and Krull [23]: that a totally ordered non-Archimedean field has a natural valuation. It is also clear that a totally ordered non-Archimedean integral domain has a natural valuation sorts. These valuations, although derived from the total order on these domains, can be defined for integral domains that are not ordered; thus an attempt at an application of this valuation theory to the residue class domains of the ring of continuous complex valued functions is natural. The solution of certain integral algebraic equations in these domains will be treated in ?1, together with the proof that these domains are integrally closed in their quotient fields (Theorem 1.8). In ?2 a place-like mapping will be defined on these domains, which has many of the algebraic (Theorem 2.2) as well as the topological (Theorem 2.5) properties of a place. In ?3 an abstract discussion will be given ofAbelian groups and integral domains with valuation, followed by an application of these ideas to the natural valuation on the residue class domains of these real and complex ftunction algebras (Theorem 3.6). It will be shown that these valuations are given by the place-like mappings defined in ?2 and can be used to analyze the structure of the prime ideals in these domains in terms of the order and algebraic properties of the value semigroup. In ?4 order properties akin to that of being an inZ-set will be studied and will be shown to be inherited, in modified form, by the value set of an Abelian group with valuation from the group. Application will then be made to the domains in question (Theorem 4.4).