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On Rayner structures
Abstract
In this note, we study substructures of generalized power series fields induced by families of well-ordered subsets of the group of exponents. We characterize the set-theoretic and algebraic properties of the induced substructures in terms of conditions on the families. We extend the work of Rayner by giving both necessary and sufficient conditions to obtain truncation closed subgroups, subrings and subfields.
... Refining the Noetherianity constraint on the support, as imposing a bound on cardinality, results in various other algebras of series, often notably motivated by the need to define exponential operators (cfr [15], [14]). Generally one can define spaces of restricted series k(Γ, F) consisting of functions f : Γ → k whose support lies in the specified ideal F of Noetherian subsets of Γ (compare with [2, Definition 6.2] and [14]). ...
... Refining the Noetherianity constraint on the support, as imposing a bound on cardinality, results in various other algebras of series, often notably motivated by the need to define exponential operators (cfr [15], [14]). Generally one can define spaces of restricted series k(Γ, F) consisting of functions f : Γ → k whose support lies in the specified ideal F of Noetherian subsets of Γ (compare with [2, Definition 6.2] and [14]). ...
... BΣVect ⊆ KTVect s ΣVect and that whether the first inclusion is an equality is a question left open (although we expect it to be strict). The inclusion BΣVect k ⊆ KTVect k,s in particular implies that in the usual context of generalized series (in particular Hahn fields, or more generally, Reyner structures, cfr [14]) preserving infinite sums can be understood in terms of continuity with respect to certain linear topologies. The separating example proving that the inclusion KTVect k,s ΣVect k is strict can already be presented here and may be an excuse to introduce a somewhat restricted version various notions in the following. ...
We discuss possible definitions of categories of vector spaces enriched with a notion of formal infinite linear combination in the likes of the formal infinite linear combinations one has in the context of generalized power series, we call these \textit{categories of reasonable strong vector spaces} (r.s.v.s.). We show that, in a precise sense, the more general possible definition for a strong vector space is that of a small -enriched endofunctor of that is right orthogonal, for every cardinal , to the cokernel of the canonical inclusion of the -th copower in the -th power of the identity functor: these form the objects for a universal r.s.v.s. we call . We relate this category to what could be understood to be the obvious category of strong vector spaces and to the r.s.v.s. of separated linearly topologized spaces that are generated by linearly compact spaces. We study the number of iterations of the obvious approximate reflector on needed to construct the orthogonal reflector as it relates to the problem of constructing the smallest subspace of an closed under taking infinite linear combinations containing a given linear subspace of H of X. Finally we show the natural monoidal closed structure on restricts naturally to and apply this to define an infinite-sum-sensitive notion of K\"ahler differentials for generalized power series. Most of the technical results apply to a more general class of orthogonal subcategories of and we work with that generality.
... In this paper we extend the approach of [KS22] to the study of v-Aut G and o-Aut G for a Hahn group G. We introduce the notion of (canonical) lifting property for a general Hahn group and, based on [KKS22], we prove that this is satisfied by a relatively large class of Hahn groups. We then decompose v-Aut G (resp. ...
... The main result of the section is Theorem 3.15, which provides a semidirect product decomposition of the v-Aut G of a Hahn group G satisfying the lifting property w.r.t. the skeleton. In Subsection 3.3, based on [KKS22], we introduce Rayner groups (Definition 3.18) and characterise those that satisfy the lifting property w.r.t. the skeleton (Theorem 3.19). Section 4 focusses on Hahn sums, providing a description of the group of order preserving automorphism as a group of matrices, generalising results of [Con58] and [DG97]. ...
... These generalise an analogous notion (Rayner fields) introduced by Rayner in [Ray68] in the context of Hahn fields. Further Rayner structures are introduced and studied in [KKS22]. ...
Hahn groups endowed with the canonical valuation play a fundamental role in the classification of valued abelian groups. In this paper we study the group of valuation (respectively order) preserving automorphisms of a Hahn group G. Under the assumption that G satisfies some lifting property, we prove a structure theorem decomposing the automorphism group into a semidirect product of two notable subgroups. We characterise a class of Hahn groups satisfying the aforementioned lifting property. For some special cases we provide a matrix description of the automorphism group.
... The global aim of this work is to describe distinguished Hahn fields. In [2] we focus on a description via k-hulls of F (where F is a set of supports in the value group) and by characterising the Hahn fields via properties of F. ...
... Throughout this work, let k be a field, let G be an ordered abelian group and let F be a family of well-ordered subsets of G. We denote by N the set of natural numbers with 0. In most other regards, we follow the notation and terminology of [2]. ...
... Following the notation in [2], we use the convention that for any wellordered subset A ⊆ G we denote by k((A)) all elements from k((G)) whose support lies in A. (ii) If F consists of all well-ordered subsets of G ≥0 , then we obtain S F = k G ≥0 . (iii) Let κ be an uncountable cardinal. ...
Within the field of Laurent series with rational coefficients, those series representing rational functions can be characterised via linear recurrence relations of their coefficients. Inspired by this result, we establish a notion of generalised linear recurrence relations for fields of generalised power series. We study distinguished Hahn fields that are determined by generalised linear recurrence relations and apply our results to the study of automorphism groups of such fields.
... Appendix A contains results from the joint work [KKS21] with L. S. Krapp and S. Kuhlmann. It consists of a further analysis of the conditions defining a Rayner group (resp. ...
... Rayner groups generalise the analogous notion of Rayner fields, introduced by Rayner in [Ray68] and which we will study in Subsection 3.3.6. Further generalisations (Rayner structures) will be studied in Appendix A, based on the joint work [KKS21] with L. S. Krapp and S. Kuhlmann. ...
... For a deeper analysis of Rayner structures see[KKS21], also summarised in Appendix A. ...
... The fields of generalized series E ⊆ K((M)) for which the inherited summability structure is more relevant are obtained by restricting the family of allowed supports in the definition of Hahn field from all well ordered sets to some suitable ideal B of subsets of M (cf [1], [7]), such subfields will be denoted by K((M)) B . These are the fields for which also the two questions above seem more relevant. ...
Let T be the theory of an o-minimal field and a common reduct of T and . I adapt Mourgues' and Ressayre's constructions to deduce structure results for -reducts of T--spherical completion of models of . These in particular entail that whenever is a reduct of defining the exponential, every elementary extension of has an elementary truncation-closed embedding in . This partially answers a question in [3](arXiv:2002.07739). The main technical result is that certain expansion of Hahn fields by generalized power series interpreted as functions defined on the positive infinitesimal elements, have the property that truncation closed subsets generate truncation closed substructures. This leaves room for possible generalizations to the case in which is power bounded but not necessarily a reduct of .
... on G(181,182), and Krapp, Kuhlmann and Serra(117) took into account several variations of axioms upon J . Both assessments dealt exclusively with the case in which the ordered group G is commutative. ...
Resumo Rayner rngs are rngs (rings without unity) whose elements are formal power series, where the coefficients lie in a rng and the exponents lie in an additive ordered group, such that the support belongs to a predetermined ideal constrained by a set of axioms. The work presents an inspection of the interplay between the algebraic, topological and categorical properties of the Rayner rngs, the rngs of coefficients and the ordered groups of exponents, studying the Rayner rngs under varied theoretical perspectives and seeking universal relations between them. Two key topologies on these structures are systematically analysed, the so-called weak and strong topologies, and a version of the Intermediate Value Theorem is obtained for the weak topology. Special attention is given to rngs of Levi-Civita, Puiseux and Hahn series, which are prominent instances of Rayner rngs.
Let k be a field, G a totally ordered abelian group. The maximal field of generalised power series , endowed with the canonical valuation v, plays a fundamental role in the classification of valued fields (Kaplansky, 1942 and 1945). In this paper we study the group of valuation preserving automorphisms of any subfield k(G)⊆K⊆K, where k(G) is the fraction field of the group ring k[G]. Under the assumption that K satisfies two lifting properties we prove a structure theorem decomposing into a 4-factor semi-direct product of notable subgroups. We identify a large class of fields satisfying the two aforementioned lifting properties. We focus on the group of strongly additive automorphisms of K. We give an explicit description of the group of strongly additive internal automorphisms in terms of the groups of homomorphisms Hom(G,k×) of G into k× and Hom(G,1+IK) of G into the group of 1-units of the valuation ring of K. To illustrate the power of our methods, we apply our results to some special cases, such as the field of Laurent series (Schilling, 1944) and that of Puiseux series (Deschamps, 2005).
Inspired by Conway's surreal numbers, we study real closed fields whose value group is isomorphic to the additive reduct of the field. We call such fields omega-fields and we prove that any omega-field of bounded Hahn series with real coefficients admits an exponential function making it into a model of the theory of the real exponential field. We also consider relative versions with more general coefficient fields.
Das Studium der nichtarchimedischen Größensysteme geht zurück auf P. Du Bois-Reymond¹ und O. Stolz.² Ausführlich finden sich einige hieher gehörende Fragen behandelt in der von der Accademia dei Lincei preisgekrönten Schrift von R. Bettazzi: Teoria delle grandezze.³ In seinen mathematisch und philosophisch bedeutungsvollen »Fondamenti di geometria«⁴ baute sodann G. Veronese eine Geometrie auf ohne Benützung des archimedischen Axioms und kam spôter, anlôßlich verschiedener gegen sein Werk gerichteter Einwônde, wiederholt auf den Gegenstand zurück.⁵ Weitere Untersuchungen über nichtarchimedische Größensysteme rühren von T. Levi-Civita her,⁶ der sich eine arithmetische Darstellung des Veronese’schen Kontinuums zum Ziele setzte und dabei zu Resultaten von großer Allgemeinheit geführt wurde. Ferner sei hier noch eine Arbeit von A. Schoenflies genannt,⁷ von der weiter unten eingehender zu sprechen sein wird.
In (K-K-S) it was shown that fields of generalized power series cannot admit an exponential function. In this paper, we construct fields of generalized power series with bounded support which admit an exponential. We give a natural definition of an exponential, which makes these fields into models of real exponentiation. The method allows to construct for everyregular uncountable cardinal, 2� pairwise non-isomorphic models of real exponenti- ation (of cardinality �), but all isomorphic as ordered fields. Indeed, the 2� exponentials constructed have pairwise distinct growth rates. This method relies on constructing lexicographic chains with many automorphisms.
We study necessary and sufficient conditions for a valued field \KF with
value group G and residue field \kf (with char \KF = char \kf) to admit
a truncation closed embedding in the field of generalized power series \kf((G,
f)) (with factor set f). We show that this is equivalent to the existence of
a family ({\it tower of complements}) of \kf-subspaces of \KF which are
complements of the (possibly fractional) ideals of the valuation ring. If \KF
is a Henselian field of characteristic 0 or, more generally, an algebraically
maximal Kaplansky field, we give an intrinsic construction of such a family
which does not rely on a given truncation closed embedding. We also show that
towers of complements and truncation closed embeddings can be extended from an
arbitrary field to at least one of its maximal immediate extensions.
Let k be any algebraically closed field, and denote by k (( t )) the field of formal power series in one indeterminate t over k . Let
so that K is the field of Puiseux expansions with coefficients in k (each element of K is a formal power series in t l/r for some positive integer r ). It is well-known that K is algebraically closed if and only if k is of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous to K which is algebraically closed when k has non-zero characteristic p . In this paper, I prove that the set L of all formal power series of the form Σ a i t ei (where ( e i ) is well-ordered, e i = m i | np rt , n ∈ Ζ, m i ∈ Ζ, a i ∈ k , r i ∈ Ν) forms an algebraically closed field.
Let us call an integer part of an ordered field any subring such that every element of the field lies at distance less than 1 from a unique element of the ring. We show that every real closed field has an integer part.