On Rayner structures
aa r X i v : . [ m a t h . A C ] A p r ON RAYNER STRUCTURES
LOTHAR SEBASTIAN KRAPP, SALMA KUHLMANN, AND MICHELE SERRA7 April 2020
Abstract.
In this note, we study substructures of generalised powerseries fields induced by families of well-ordered subsets of the groupof exponents. We relate set theoretic and algebraic properties of thefamilies to algebraic features of the induced sets. By this, we extendthe work of Rayner [8] to truncation closed substructures of generalisedpower series fields. Introduction
In his work [8], Rayner presented a construction method for subfields ofgeneralised power series fields induced by families of well-ordered subsets ofthe value group. More specifically, for a field k , an ordered abelian group G and a family F of well-ordered subsets of G , Rayner introduced sufficientconditions on F in order that the k -hull of F (Definition 2 .
4) forms a subfieldof k (( G )). Here k (( G )) denotes the field of generalised power series withcoefficient field k and value group G .In this note, we study necessary and sufficient conditions on the field k , the group G and the family F , for the k -hull of F to satisfy certainproperties. By a careful analysis of these conditions, we characterise whenthe k -hull of F is a subgroup (Proposition 3 . . .
10) and a subfield (Proposition 3 .
12) of k (( G )). Amongthese, Hahn fields (Definition 2 .
3) are of special interest. In particular,Mourgues and Ressayre [6] studied the interesting class of truncation closedHahn fields. In Corollary 3 .
13, we characterise k -hulls that are truncationclosed Hahn fields. Finally, we show in Theorem 3 .
15 that the k -hull of F is a Rayner field (Definition 3 .
2) if and only if it is a Hahn field.2.
Preliminaries, terminology and notations
Throughout this note, let k be a field, let G be an additiveordered abelian group and let F be a family of well-ordered subsetsof G . For
A, B ⊆ G and g ∈ G we denote by W ( A ) the family of well-orderedsubsets of A , by h A i the subgroup of G generated by A , by A ⊕ B the setof sums { a + b | a ∈ A, b ∈ B } , by A + g the translation set { a + g | a ∈ A } and by L n ∈ N A the set of finite sums of elements of A , i.e. L n ∈ N A = Mathematics Subject Classification : 13J05 (12J20 16W60 06F20) { P ni =1 a i | n ∈ N , a , . . . , a n ∈ A } (where N = { , , . . . } ). By convention, L n ∈ N ∅ = { } .On F we consider the following: Conditions 2.1. (i)
F 6 = ∅ ;(ii) { } ∈ F ;(iii) g ∈ G implies { g } ∈ F ;(iv) A ∈ F and B ⊆ A implies B ∈ F ;(v) if A ∈ F and B is an initial segment of A , then B ∈ F ;(vi) A, B ∈ F implies A ∪ B ∈ F ;(vii) A, B ∈ F implies A ⊕ B ∈ F ;(viii) A ∈ F and A ⊆ G ≥ implies L n ∈ N A ∈ F ;(ix) if g ∈ G such that { g } ∈ F , then {− g } ∈ F ;(x) (cid:10)S A ∈F A (cid:11) = G ;(xi) A ∈ F and g ∈ G implies A + g ∈ F . Remark 2.2.
Some of the Conditions 2 . k andvalue group G by K . It consists of the set of all functions s : G → k whose support supp( s ) = { g ∈ G | s ( g ) = 0 } is a well-ordered subset of G . Forany g ∈ G , we denote by t g the characteristic function mapping g to 1 andeverything else to 0, and we call t g a (monic) monomial of K . This way, wecan express a power series s ∈ K by s = P g ∈ G s g t g , where s g = s ( g ) ∈ k . Forany power series r, s ∈ K , their sum is given by r + s = P g ∈ G ( r g + s g ) t g andtheir product by rs = P g ∈ G c g t g with c g = P h ∈ G r h s g − h . These operationsmake K a field (cf. [4, 7]). Definition 2.3.
We call K the maximal Hahn field with coefficient field k and value group G . A subfield K of K with { αt g | α ∈ k, g ∈ G } ⊆ K iscalled a Hahn field in K .Now we introduce the subsets of K induced by F that we are going tostudy. Definition 2.4.
We call the set k (( F )) = { a ∈ K | supp( a ) ∈ F } ⊆ K the k -hull of F in K . Remark 2.5.
Note that k (( F )) contains the coefficient field k if and only if ∅ ∈ F and { } ∈ F . Notation 2.6.
Whenever the family F is of the form W ( S ) for some set S ⊆ G , we write k (( S )) instead of k (( F )) . Example 2.7. (1) In the valuation theoretic study of maximal Hahn fields, k -hulls play an important role, as they give rise to the valuation ring and N RAYNER STRUCTURES 3 its maximal ideal: for the standard valuation v min : K \ { } → G, s min supp( s ) on K , the valuation ring is given by k (cid:0)(cid:0) G ≥ (cid:1)(cid:1) and its max-imal ideal by k (cid:0)(cid:0) G > (cid:1)(cid:1) .(2) Let κ be an uncountable regular cardinal and let F be the family of allwell-ordered subsets of G of cardinality less than κ . Then k (( F )) is theHahn field of κ -bounded power series. Such Hahn fields provide naturalconstructions for models of real exponentiation (cf. [5, 1]).We lastly introduce the notions of restriction and truncation closure for k -hulls. Due to the work of Mourgues and Ressayre [6], truncation closedsubfields of maximal Hahn fields are of particular interest in the study ofinteger parts of ordered fields and have been the subject of study ever since(cf. e.g. [2, 3]). Definition 2.8.
The k -hull k (( F )) of F is called restriction closed if F satisfies (iv). It is called truncation closed if F satisfies (v).3. Properties of k -hulls We start by summarising the sufficient conditions on F given in [8] inorder to ensure that k (( F )) has certain algebraic properties as the followingtheorem (cf. [8, page 147]). Theorem 3.1. (1) If F satisfies (i) , (iv) and (vi) , then k (( F )) is a sub-group of ( K , +) .(2) If F satisfies (i) , (iv) and (vi) , (viii) and (xi) , then k (( F )) is asubring (with identity) of K .(3) If F satisfies (i) , (iv) and (vi) , (viii) , (x) and (xi) , then k (( F )) is asubfield of K . Theorem 3 . Definition 3.2.
We call F a Rayner field family in G if it satisfies con-ditions (i), (iv), (vi), (viii), (x) and (xi). If F is a Rayner field family in G , then we call the field k (( F )) a Rayner field (with coefficient field k andfield family F ). Remark 3.3.
Rayner does actually not include (i) in his definition of afield family. However, if G = { } , then the empty family would satisfy (iv),(vi), (viii), (x) and (xi) but its k -hull would be the empty set. In fact by ourdefinition, if G = { } , then the only Rayner field family in G is F = {∅ , { }} .Since Rayner is merely interested in sufficient conditions on F in order toensure that k (( F )) exhibits certain algebraic properties, some of the condi-tions he poses may not be necessary. In the following, we carefully analysefurther the relations between the Conditions 2 . k (( F ))as an algebraic substructure of K . Proposition 3.4.
Suppose that k = F . Then k (( F )) is a subgroup of ( K , +) if and only if F satisfies (i) , (iv) and (vi) . L. S. KRAPP, S. KUHLMANN, AND M. SERRA
Proof.
Theorem 3 . k (( F )) is an additive subgroup of K . Then it contains 0, whence ∅ =supp(0) ∈ F . This establishes (i). Now let A ∈ F and let B ⊆ A . Let a besuch that supp( a ) = A and let c ∈ K be defined by c g / ∈ { , − a g } , g ∈ B ; (which is possible as k = F ) c g = − a g , g ∈ A \ B ; c g = 0 , g ∈ G \ A. Then supp( c ) = A , whence c ∈ k (( F )). Since k (( F )) is a group, we have a + c ∈ k (( F )) and thus B = supp( a + c ) ∈ F , yielding (iv). Now let A, B ∈ F and let a ∈ K be such that supp( a ) = A . Then choose b ∈ K suchthat supp( b ) = B and b g = − a g for every g ∈ A ∩ B (this is always possiblesince k = F ). This yields supp( a + b ) = A ∪ B ∈ F and thus establishes(vi). (cid:3) Corollary 3.5.
Suppose that k = F . If k (( F )) is an additive group, then itis restriction closed (and thus, in particular, truncation closed). We now show that the conclusion of Proposition 3 . k = F .Note that for any family F , there is a bijective correspondence between F and F (( F )) given by A P g ∈ A t g . Example 3.6.
Let F ( s ) be the subfield of F (( Z )) generated by s = t + t .We show that F ( s ) does not contain t and is thus not truncation closed. Itsuffices to prove that for any p, q ∈ F [ X ] with q = 0 we have t q ( s ) = p ( s ).We do so by induction on the degree of p .Clearly, if deg( p ) = 0, then for any q ∈ F [ X ] we have t q ( s ) = p ( s ). Let n ∈ N and suppose that the claim holds for any polynomial of degree n − p ( X ) = X n + P n − i =0 a i X i and assume, for a contradiction, that for some q ∈ F [ X ] \ { } we have t q ( s ) = p ( s ). Let q ( X ) = P mj =0 b j X j . Then m X j =0 b j t ( t + t ) j = t q ( s ) = p ( s ) = ( t + t ) n + n − X i =0 a i ( t + t ) i . Comparing coefficients of t , we obtain a = 0 and thus m X j =0 b j t ( t + t ) j = ( t + t ) ( t + t ) n − + n − X i =1 a i ( t + t ) i − ! . Comparing coefficients of t , we obtain a = 0, whence m X j =0 b j t ( t + t ) j = ( t + t ) ( t + t ) n − + n − X i =2 a i ( t + t ) i − ! . Finally, comparing coefficients of t , we obtain b = 0. Hence, t m − X j =0 b j +1 s j = s s n − + n − X i =2 a i s i − ! . N RAYNER STRUCTURES 5
This shows that for the polynomial p ′ ( X ) = X (cid:16) X n − + P n − i =2 a i X i − (cid:17) ofdegree n − q ′ ( X ) such that t q ′ ( s ) = p ′ ( s ) giving us the requiredcontradiction. Now let F be the set of all supports of elements of F ( s ). Then F ( s ) = F (( F )) and F (( F )) is a subfield of F (( Z )) which is not truncationclosed. (cid:3) We now also consider multiplication on k (( F )). Lemma 3.7. If F satisfies (i) , (iv) , (vi) and (vii) , then k (( F )) is a subring(possibly without identity) of K .Proof. By Proposition 3 . k (( F )) is an additive subgroup of ( K , +). Nowlet a, b ∈ k (( F )). We set A = supp( a ), B = supp( b ) and let c = ab ∈ K .Then by definition of the product, we have supp( ab ) ⊆ A ⊕ B ∈ F . Hence,by (vii) we obtain supp( ab ) ∈ F and thus ab ∈ k (( F )). (cid:3) Proposition 3.8.
Suppose that char( k ) = 0 . Then k (( F )) is a subring(possibly without identity) of K if and only if F satisfies conditions (i) , (iv) , (vi) and (vii) .Proof. The backward direction follows from Lemma 3 .
7. For the converse,suppose that k (( F )) is a subring of K . By Proposition 3 .
4, it remains toverify (vii). Let
A, B ∈ F and set a = P g ∈ A t g and b = P g ∈ B t g . Thensince char( k ) = 0, we obtain that A ⊕ B = supp( ab ) ∈ F . (cid:3) Corollary 3.9.
Suppose that char( k ) = 0 . Then whenever k (( F )) is a sub-ring of K , it is restriction closed (and, in particular, truncation closed). The condition char( k ) = 0 in Proposition 3 . a and b do notcancel in the product ab , whence supp( ab ) = supp( a ) ⊕ supp( b ). This canalso be ensured by a condition on the cardinality of k as the followig resultshows. Proposition 3.10.
Suppose that | k | > | G | . Then k (( F )) is a subring (pos-sibly without identity) of K if and only if F satisfies conditions (i) , (iv) , (vi) and (vii) .Proof. Again, the backward direction follows from Lemma 3 .
7, and for theconverse, if k (( F )) is a subring of K , by Proposition 3 . A, B ∈ F . Since A and B are well-ordered, we can let α and β bethe ordinals respresenting the order type of A and B , respectively. Morever,we can enumerate A and B by A = { a γ | γ < α } and B = { b γ | γ < β } . Wenow construct c, d ∈ k (( F )) with supports A and B , respectively, such thatsupp( cd ) = A ⊕ B . Set d = P g ∈ B t g . Then for any h ∈ A ⊕ B we have that(3.1) ( cd ) h = X γ<αh − a γ ∈ B c a γ . L. S. KRAPP, S. KUHLMANN, AND M. SERRA
We define c a γ for γ < α inductively. Set c a = 1. Now suppose that forsome κ < α , we have already constructed c a γ for each γ < κ . Then let c a κ ∈ k \ { } be such that c a κ is not equal to the negative of any finitesum of elements from { c a γ | γ < κ } . This is possible, as | k | > | G | . Nowlet h ∈ A ⊕ B and let µ be the largest ordinal such that c a µ appears inthe expression of ( cd ) h given in (3.1). Then ( cd ) h − c a µ is a finite sum ofelements from { c a γ | γ < µ } . By the construction of the c a γ , we obtain( cd ) h − c µ = − c µ and thus ( cd ) h = 0, as required. (cid:3) We now consider the field structure on k -hulls. Lemma 3.11.
Suppose that F satisfies conditions (ii) , (iv) , (vi) , (vii) , (viii) and (ix) . Then k (( F )) is a subfield of K .Proof. Proposition 3 . . k (( F )) is a ring withidentity. Let b ∈ k (( F )) \ { } be arbitrary and let h = min supp( b ). Then by(iv) and (ix), we have t − h ∈ k (( F )) and thus obtain b − h t − h b = 1 + X g ∈ G > b − h b g t g − h ∈ k (( F )) . Now set a = − P g ∈ G > b − h b g t g − h and let A = supp( a ) ⊆ G > ∈ F . Then(1 − a ) − = P ∞ i =0 a i (cf. Neumann’s Lemma [7, page 211]). Hence, supp(1 − a ) − ⊆ L n ∈ N A and, by (viii) and (iv), it lies in F . This implies b h t h b − = (cid:0) b − h t − h b (cid:1) − ∈ k (( F )), whence b − ∈ k (( F )), as required. (cid:3) Proposition 3.12.
Suppose that char( k ) = 0 . Then k (( F )) is a subfield of K if and only if F satisfies conditions (ii) , (iv) , (vi) , (vii) , (viii) and (ix) .Proof. By Lemma 3 .
11, only the forward direction needs to be shown. Let k (( F )) be a field. Then Proposition 3 . . F satisfyconditions (ii), (iv), (vi) and (vii). To prove condition (viii) let A ∈ F besuch that A ⊆ G ≥ and let a = P g ∈ A > t g . Then supp(1 − a ) = A . ByNeumann’s Lemma, (1 − a ) − = P ∞ i =0 a i . As char( k ) = 0, the support of(1 − a ) − is L n ∈ N A . Since k (( F )) is a field, (1 − a ) − ∈ k (( F )), establishing(viii). Finally, (ix) follows easily, as for any monomial t g ∈ k (( F )) we alreadyhave t − g ∈ k (( F )). (cid:3) As a corollary, we obtain necessary and sufficient conditions (in the casechar( k ) = 0) in order that k (( F )) is a Hahn field. Corollary 3.13.
Suppose that char( k ) = 0 . Then k (( F )) is a Hahn field in K if and only if F satisfies conditions (iii) , (iv) , (vi) , (vii) , (viii) . Hence,if k (( F )) is a Hahn field in K , then it is restriction closed and thus alsotruncation closed.Proof. If k (( F )) is a Hahn field, then F clearly satisfies (iii). The otherproperties follow from Proposition 3 .
12. For the converse, note that (iii)implies (ii) and (ix). The rest follows from Proposition 3 . (cid:3) N RAYNER STRUCTURES 7
Finally, we show that k (( F )) is a Hahn field if and only if it is a Raynerfield. By Corollary 3 .
13, it suffices to show that F is a Rayner field familyif and only if it satisfies (iii), (iv), (vi), (vii), (viii). We first prove that if G is non-trivial, then condition (x) in Definition 3 . Lemma 3.14.
Suppose that G = { } and that F satisfies conditions (iv) , (viii) and (xi) . Then (i) ⇐⇒ (ii) ⇐⇒ (iii) ⇐⇒ (x) . Proof. (i) ⇒ (ii): Let F 6 = ∅ and let A ∈ F . If A = ∅ , then for any g ∈ A ,we obtain by (iv) and (xi) that { } = { g } − g ∈ F . Otherwise, by (viii) wehave L n ∈ N A = { } ∈ F . (ii) ⇒ (iii): This follows immediately from (xi).(iii) ⇒ (x) and (x) ⇒ (i) are obvious. Note that for the latter we need that G = { } . (cid:3) Theorem 3.15.
Suppose that char( k ) = 0 . Then k (( F )) is a Rayner field ifand only if it is a Hahn field.Proof. Suppose that k (( F )) is a Rayner field, that is, F satisfies (i), (iv),(vi), (viii), (x) and (xi). By Corollary 3 .
13, it remains to verify (iii) and(vii). If G = { } , then by Remark 3 . F = {∅ , { }} , which triviallysatisfies (iii) and (vii). If G = { } , then Lemma 3 .
14 shows that F satisfies(iii). We thus only have to show (vii). Let A, B ∈ F be non-empty. Let a = min A and b = min B . Then by (xi), we have A − a, B − b ∈ F . Notethat A − a, B − b ∈ G ≥ . Hence, by (vi) and (viii), we obtain M n ∈ N (( A − a ) ∪ ( B − b )) ∈ F . In particular, ( A − a ) ⊕ ( B − b ) ∈ F . By (xi) we obtain A ⊕ B = (( A − a ) ⊕ ( B − b )) + ( a + b ) ∈ F .Vice versa, suppose that k (( F )) is a Hahn field, that is, F satisfies (iii),(iv), (vi), (vii), (viii). We need to show that (i), (x) and (xi) hold. Again,if G = { } , then F = {∅ , { }} and there is nothing to prove. Otherwise, byLemma 3 .
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Fachbereich Mathematik und Statistik, Universit¨at Konstanz, 78457 Kon-stanz, Germany
E-mail address : [email protected] Fachbereich Mathematik und Statistik, Universit¨at Konstanz, 78457 Kon-stanz, Germany
E-mail address : [email protected] Fachbereich Mathematik und Statistik, Universit¨at Konstanz, 78457 Kon-stanz, Germany
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