The hyperserial field of surreal numbers

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Preprint submitted on 22 May 2021

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Vincent Bagayoko, Joris van der Hoeven

To cite this version:

Vincent Bagayoko, Joris van der Hoeven. The hyperserial field of surreal numbers. 2021. �hal- 03232836�

Vincent Bagayoko^A,UMons, LIX

Joris van der Hoeven^B,CNRS, LIX

For any ordinal α>0, we show how to define a hyperexponential E_ω^α and a hyperlogarithmL_ω^α on theclassNo^>,≻of positive infinitely large surreal numbers. Suchfunctions are archetypes ofextremely fast and slowlygrowing functions at infinity. We also show that the surreal numbersforma so-called hyperserialfieldfor our definition.

1 Introduction

The orderedfieldNoofsurreal numbers was introducedby Conway in[11].Conway originally used transfinite recursion to define both the surreal numbers(henceforthcallednumbers), the orderingonNo, and the ringoperations. For any two setsLandRofnumbers withL<R(i.e.

x<yfor allx∈Landy∈R), there exists a number{L∣R}with L < {L∣R} < R,

and all numberscanbe obtained in this way. Givenx={xL∣x_R}andy={yL∣y_R}, we have x+y ≔ {xL+y,x+yL∣xR+y,x+yR}

and similar recursiveformulas existfor−x,x yandfor decidingwhetherx=y,x⩽y, andx<y.

It is truly remarkable thatNoturns out tobe a totally ordered real-closedfieldfor such“simple”

definitions [11]. The bracket{∣ }iscalled theConway bracket. Using thisbracket, we obtain a surreal number in any traditional Dedekindcut, which allows us to embedℝ into No. In addition,Nocontains all ordinal numbers

0={∣}, 1={0∣}, 2={0,1∣}, . . . , ω={0,1,2, . . . ∣}, ω+1={0,1,2, . . . ,ω∣}, . . . , soNois actually aproperclass.

An interesting question is which other operationsfrom calculuscanbe extended to the sur- real numbers. Gonshor has shown how to extend the real exponentialfunction to the surreal numbers[19]and the resultingexponentialfield(No,exp)turns out tobe elementarily equiva- lent to(ℝ,exp) [13]. Berarducci and Mantova recently defined a derivation with respect toωon the surreals[9], again withgoodmodel-theoretic properties[2].Incollaboration with Mantova, the authorsconstructed a surreal solution to thefunctional equation

Eω(x+1) = expEωx,

which is abijection ofNo^>,≻≔{x∈No:x> ℝ}onto itself [6]. Wecall Eω ahyperexponential and itsfunctional inverseLωahyperlogarithm.

A. [email protected] B. [email protected]

∗. This article hasbeen writtenusing GNUTEX_MACS[27].

Thefirstgoal ofthispaper is to extend the resultsfrom [6]to theconstruction ofhyperexpo- nentialsEω^α:No^>,≻⟶No^>,≻ofany ordinalforce α, together with theirfunctional inversesLω^α. If α=β+1is a successor ordinal, thenE_ω^α satisfies thefunctional equation

E_ωβ+1(x+1) = E_ωβ(E_ωβ+1(x)).

Our secondgoal is to show that these hyperexponentials are“well-behaved”in the sense that they endowNowith the structure ofahyperserial field in the sense of [5].

1.1 Motivationand background

Whereas it is natural to study surreal exponentiation and differentiation, it may seem more exoticto define and investigate theproperties ofsurreal hyperexponentials and hyperlogarithms.

Infact, themainmotivationbehind our work is aconjecture by the second author [26,p. 16]

and a researchprogramthat was laid out in[1] forprovingthisconjecture. Theultimategoal is to expose the deep connectionsbetween two types of mathematical infinities:numerical infini- ties andgrowth rates at infinity. Letusbriefly recall the rationalebehind thisconnection.

Cantor's ordinal numbersprovide us with a way to countbeyond all natural numbers and tokeep counting beyond the size ofany set. However, ordinal arithmeticis ratherpoor in the sense that we have no subtraction or division and that addition andmultiplication do not satisfy theusual laws ofarithmetic, such ascommutativity. Wemay regard Conway's surreal numbers as providing acalculus with Cantor's ordinal numbers which does extend theusualcalculus with real numbers. In this sense, Conwaymanaged toconstruct theultimate framework for computations with numerical infinities.

Another source for computations with infinitely large quantities stems from the study of growth rates ofrealfunctions at infinity. Thefirstmajor results towards a systematicasymptotic calculus ofthiskind are due toHardy in[21,22],based on earlier ideasby du Bois-Reymond[15, 16,17]. Hardy defined an L-functiontobe a functionconstructed from x and the real num- bersℝusingthe field operations, exponentiation, and logarithms. Heproved that the germs of L-functions at infinity form a totally orderedfield. Theframework of L-functions is suit- ablefor asymptoticanalysis since we have an ordering for comparing the growth at infinity ofany two suchfunctions. This is oſten rephrasedby sayingthatL-functions have aregular growth at infinity.

Hardy also observed [21, p. 22] that “The only scales of infinity that are of any practical importance in analysis are those whichmay beconstructed bymeans of the logarithmicand exponential functions.” In other words,Hardy suggested that the framework of L-functions not only allowsfor the development ofa systematicasymptotic calculus,but that thisframe- work is also sufficient for all“practical” purposes. Alas, there are several “holes”. First ofall, the frameworkis notclosedundervarioususeful operations such as functional inversion and integration.Secondly, theframeworkdoes notcontain anyfunctions ofextremelyfast or slow growth at infinity, likeE_ω andL_ω, although suchfunctions naturally appear in the analysis of certain algorithms.For instance, thebestknown algorithm formultiplyingtwopolynomials of degreenin𝔽₂[x]runs in timeO(nlogn4^Lωn);see[23].

This raises thequestion how toconstruct a trulyuniversalframework forcomputations with regular functions at infinity. Our next candidate is the class of transseries. A transseries is aformal object that isconstructedfromx(withx→ ∞)and the real numbers,usingexponenti- ation, logarithms, andinfinitesums. One example ofa transseries is

e^{ex+ex/2+ex/3+⋅ ⋅ ⋅}−3e^x2+5 (logx)^π+42+x⁻¹+2x⁻²+6x⁻³+24x⁻⁴+ ⋅ ⋅ ⋅ + e^−x.

Dependingonconditions satisfiedby their supports, there are different types oftransseries. The firstconstructions of fields oftransseries are due toDahn andGöring [12]andÉcalle[18].More generalconstructions wereproposed subsequentlyby the second author and hisformer student Schmeling [24,25,29].Clearly, anyL-function is a transseries,but theclass oftransseries is also closedunder integration andfunctional inversion,contrary to theclass of L-functions.

However, theclass oftransseries still does notcontain any hyperexponential or hyperloga- rithmicelements likeE_ωxorL_ωx. In ourquestfor a trulyuniversalframework for asymptotic analysis, we are thus lead to look beyond: ahyperseries is aformal object that is constructed fromxand the real numbersusingexponentiation, logarithms, infinite sums, as well as hyper- exponentialsE_ω^α and hyperlogarithmsL_ω^αofanyforceα. The hyperexponentialsE_ω^α and the hyperlogarithmsL_ω^α are required to satisfyfunctional equations

E_ω^α+1∘T1 = Eω^α∘E_ω^α+1 (1.1)

L_ω^α+1∘Lω^α = T₋₁∘L_ω^α+1, (1.2)

whereTs(u)≔u+s.Forγ= ∑_i=1^p ω^αiniin Cantor normalformwithα₁< ⋅⋅⋅ <αp, we also define Lγ = L_ω∘n^α11∘ ⋅ ⋅ ⋅ ∘L_ω∘nαpp (1.3) and we require that

Lγ′ = 1

∏_β<γL_β. (1.4)

It is non-trivial to construct fields ofhyperseries in which these and several other technical properties (see section 4 below) are satisfied. This was first accomplished by Schmeling for hyperexponentialsEωⁿ and hyperlogarithms Lωⁿof finiteforce n∈ ℕ. The generalcase was tackled in[14,5].

Theconstruction of general hyperseries relies on the definition ofan abstract notion ofhyper- serial fields. Whereas the hyperseries that we are really aſter should actuallybe hyperseriesin an infinitely large variablex, abstract hyperserial fields potentially contain hyperseries that can notbe written as infinite expressions inx. In thepresentpaper, we define hyperexponen- tialsEω^αand hyperlogarithmsLω^αonNofor all ordinalsαand show that thisprovidesNowith the structure ofan abstract hyperserialfield. Moreover, any hyperseriesf inx can naturally be evaluated atx=ω toproduce a surreal numberf(ω). Theconjecturefrom [26,p. 16]states that,for a sufficientlygeneral notion of “hyperseries inx”,all surreal numberscan actuallybe obtained in this way. Weplan toprove this and theconjecture in afollow-up paper.

1.2 Generaloverviewandsummaryof ournew contributions

Ourmaingoal is to define hyperexponentialsEω^α:No^>,≻⟶No^>,≻for any ordinalα>1and to show thatNois a hyperserialfieldfor these hyperexponentials. Since ourconstructionbuilds onquite someprevious work, thepaper starts with three sections ofreminders.

In section2, we recallbasic facts about well-based series and surreal numbers. Inparticular, we recall that any surreal numberx∈Nocanbe regarded as a well-based series

x =

𝔪∈Mo

x𝔪𝔪

with real coefficientsx_𝔪∈ ℝ. The corresponding group of monomials Mo consists of those positive numbers𝔪 ∈No^>that are oftheform𝔪 ={ℝ^>L∣ ℝ^>R} forcertain subsetsLandR ofNowithℝ^>L< ℝ^>R.

Section3is devoted to the theory ofsurreal substructuresfrom [4]. One distinctivefeature oftheclass ofsurreal numbers is that itcomes with apartial, well-founded order⊑, which is called thesimplicityrelation. The Conwaybracketcan thenbecharacterizedby thefact that,for anysets LandRofsurreal numbers withL<R, there exists aunique⊑-minimal number{L∣R}

withL<{L∣R}<R. Formany interestingsubclassesSofNo, it turns out that the restrictions of⩽and⊑toSgive rise to a structure(S, ⩽, ⊑)that is isomorphicto(No, ⩽, ⊑). SuchclassesS arecalledsurreal substructuresofNoand theycome with their own Conwaybracket{∣}_S.

In section4, we recall the definition ofhyperserial fieldsfrom [5]and themain results on how to construct suchfields. One major fact from [5] on which we heavily rely is that the construction ofhyperserialfieldscanbe reduced to theconstruction of hyperserial skeletons.In the context ofthepresentpaper, thismeans that it suffices to define the hyperlogarithms Lω^α

onlyforvery special, socalledL<ω^α-atomic elements.

In thecase when α=0, the L_<1-atomic elements are simply themonomials inMo and the definition ofthe general logarithm onNo^>indeed reduces to the definition ofthe logarithm onMo: givenx∈No^>, we writex=c𝔪(1+ε), wherec∈ ℝ,𝔪 ∈Moandε is infinitesimal, and we takelogx≔log𝔪 +logc+ε−ε²/2+ε³/3+ ⋅ ⋅ ⋅. Thisvery specialcase willbeconsidered in more detail in section5.

In thecase whenα=1, theL<ω-atomicelements ofNo^>,≻are those elements𝔞 ∈No^>,≻such thatLn𝔞is amonomialfor everyn∈ ℕ. Theconstruction of LωonNo^>,≻then reduces to the construction of Lω on the classMoω of L<ω-atomic numbers. Thisparticular case wasfirst dealt with in[6]and thispapercanbeused as an introduction to themoregeneral results in the presentpaper.

For general ordinals α, we say that 𝔞 ∈No^>,≻ is L<ω^α-atomic if L_β𝔞 is a monomial for every β<α. The advantage ofrestricting ourselves to such numbers𝔞when defininghyper- logarithms is that Lα𝔞 only needs to verify few requirements with respect to the ordering.

Thismakes itpossible to defineLα𝔞usingthefairly simple recursiveformula

Lα𝔞 ≔ {ℝ,L_α𝔞′ +(L_<α𝔞′)⁻¹∣Lα𝔞′′ −(L_<α𝔞)⁻¹,L<α𝔞}, (1.5)

where𝔞′, 𝔞′′range overL_<α-atomicnumbers with𝔞′, 𝔞′′ ⊑ 𝔞and𝔞′ < 𝔞 < 𝔞′′;see also(7.1).

In section6, weprove that this definition is warranted and that the resulting functions Lα

satisfy the axioms ofhyperserial skeletonsfrom [5, Section3]. Ourproof proceedsby induction on α and also relies on the fact that the class Moα of L<ω^α-atomic numbers actuallyforms a surreal substructure ofNo. Ourmain result is thefollowingtheorem:

Theorem 1.1. The definition (1.5) gives No the structure of a confluent hyperserial skeleton in the sense of [5]. Consequently, we may uniquely extend L_ω^μto No^>,≻in a way that givesNo the structure of a confluent hyperserial field. Moreover, for each ordinal μ, the extended function L_ω^μ:No^>,≻⟶No^>,≻is bijective.

Ourfinal section7is devoted to further identities that illustrate the interplay between the hyperexponential and hyperlogarithmic functions and the simplicity relation⊑onNo. We also prove thefollowing more symmetric variant of (1.5):

Lα𝔞 = {ℝ,L_α𝔞′ +(L_<α𝔞′)⁻¹∣Lα𝔞′′ −(L_<α𝔞′′)⁻¹,L<α𝔞}, (1.6)

where 𝔞′, 𝔞′′ again range over theL<α-atomic numbers with𝔞′, 𝔞′′ ⊑ 𝔞and 𝔞′ < 𝔞 < 𝔞′′. An interesting open question is whether there exists an easy argument that would allow us to use(1.6)instead of (1.5)as a definition of Lα𝔞.

2 Basic notions

2.1 Ordered fieldsof well-basedseries 2.1.1 Well-based series

Let(𝔐, ×,1, ≺) be a(possiblyclass-sized)linearly ordered abeliangroup. We write𝕊 ≔ ℝ[[𝔐]]

for theclass of functionsf: 𝔐 ⟶ ℝwhose support

suppf ≔ {𝔪 ∈ 𝔐 :f(𝔪)≠0}

is awell-basedset, i.e. a set which is well-ordered with respect to the reverse order(𝔐, ≻).

We see elementsf of𝕊asformalwell-based series f = ∑_𝔪f_𝔪𝔪, wheref_𝔪denotes thecoeffi- cient f(𝔪)∈ ℝof𝔪inf,for each𝔪 ∈ 𝔐.Ifsuppf ≠ ∅, then we define𝔡_f≔maxsuppf ∈ 𝔐to be thedominant monomialof f. For𝔪 ∈ 𝔐, we let f_≻𝔪≔ ∑_𝔫≻𝔪f_𝔫𝔫and we write f_≻≔f_≻1. We say that a seriesg∈ 𝕊is atruncationof f and we writeg{^{f ifsupp (f −g)≻}g. The relation{ is a well-foundedpartial order on𝕊withminimum0.

By[20], theclass𝕊is an orderedfieldunder thepointwise sum (f +g) ≔

𝔪

(f_𝔪+g_𝔪)𝔪, the Cauchyproduct

f g ≔

𝔪

(((((((((

_{𝔲𝔳=𝔪}^f𝔲g𝔳

)))))))))

^𝔪,

(where each sum∑_{𝔲𝔳=𝔪}f_𝔲g_𝔳hasfinite support), and where thepositivecone𝕊^>={f ∈𝕊:f >0}

isgivenby

𝕊^> ≔ {f ∈ 𝕊 :f ≠0∧f𝔡_f>0}.

The identification of𝔪 ∈ 𝔐 with theformal series∑_𝔫=𝔪1⋅ 𝔫 ∈ 𝕊 induces an orderedgroup embedding(𝔐, ×, ≺)⟶(𝕊^>, ×, <).

We next define thefollowingasymptoticrelations on𝕊:

f ≺g ⟺ ℝ^>|f|<|g|

f ≼g ⟺ ∃r∈ ℝ^>,|f|⩽r|g|

f ≍g ⟺ f ≼g≼f.

The relation≺extends the orderingon𝔐. For non-zerof,g∈ 𝕊we actually havef ≺g(resp.

f ≼g, resp.f ≍g)ifand only if𝔡_f≺ 𝔡g (resp.𝔡_f≼ 𝔡g, resp.𝔡_f= 𝔡g). Wefinally define 𝕊_≻ ≔ {f ∈ 𝕊 :suppf ⊆ 𝔐^≻}

𝕊^≺ ≔ {f ∈ 𝕊 :suppf ⊆ 𝔐^≺} = {f ∈ 𝕊 :f ≺1}

𝕊^>,≻ ≔ {f ∈ 𝕊 :f > ℝ}={f ∈ 𝕊 :f ⩾0∧f ≻1}.

Series in𝕊≻,𝕊^≺and𝕊^>,≻are respectivelycalledpurely large,infinitesimal, andpositive infinite.

2.1.2 Well-based families

Let(fi)i∈I be afamily in𝕊,We say that(fi)i∈I iswell-basedif i. ⋃_i∈Isuppf_iis well-based, and

ii. {i∈I: 𝔪 ∈suppfi}isfinitefor all𝔪 ∈ 𝔐.

In thatcase, wemay define the sum∑_i∈Ifiof(fi)i∈I by

i∈I

f_i ≔

𝔪

(((((((((

_i∈I ^(fi)𝔪

)))))))))

^𝔪.

If𝕌 = ℝ[[𝔑]]is anotherfield ofwell-based series andΨ: 𝕊⟶ 𝕌isℝ-linear, then we say thatΨ isstrongly linear if for every well-basedfamily(f_i)_i∈I in𝕊, thefamily(Ψ(fi))_i∈I is well-based, with

Ψ

(((((((((

_i∈I^fi

)))))))))

⁼ _i∈I^Ψ(fi).

2.2 Surrealnumbers

2.2.1 Surreal numbers and simplicity

We denotebyOntheclass ofordinal numbers.Following [19], we defineNotobe theclass of sign sequences

a = (a[β])_β<α ∈ {−1, +1}^α

ofordinallength α∈On. The termsa[β]∈{−1, +1}arecalled thesignsof aand we writelafor the length of a. Given two numbersa,b∈No, we define

a⊑b ⟺ la⩽l_b∧(∀β<la,a[β]=b[β]).

Wecall⊑thesimplicity relationonNoand note that(No, ⊑)is well-founded.See[4, Section2]

formore details about the interactionbetween⊑and the orderedfield structure ofNo.

Recall that the Conwaybracket ischaracterizedby thefact that,for anysets LandRofsurreal numbers withL<R, there exists aunique⊑-minimal number{L∣R}withL<{L∣R}<R. Con- versely,given a numbera∈No, we define

aL ≔ {x∈No:x⊏a,x<a}

aR ≔ {x∈No:x⊐a,x>a}.

Thenacancanonicallybe written as

a = {a_L∣a_R}.

2.2.2 Ordinals as surreal numbers

The structure(No, ⊑) contains an isomorphic copy of(On, <) by identifyingeach ordinalαwith the constant sequence(+1)_β<α oflength α. We will writeν⩽Onto state thatν is either an ordinal or theclass ofordinals.

Forγ∈On, we writeω^γfor the ordinal exponentiation of ωto thepowerγand we define ω^On ≔ {ω^γ:γ∈On}.

If μ∈Onis a successor ordinal, then we defineμ−tobe theunique ordinal withμ=μ−+1. We also defineμ−≔μif μis a limit ordinal.Similarly, if α=ω^μ, then we setα/ω≔ω^μ−. Recall that every ordinalγhas aunique Cantor normalform

γ = ω^η1n₁+ ⋅ ⋅ ⋅ +ω^ηrn_r, wherer∈ ℕ,n₁, . . . ,n_r∈ ℕ^>0andη₁, . . . ,η_r∈Onwithη₁> ⋅ ⋅ ⋅ >η_r. 2.2.3 Surreal numbers as well-based series

We defineMotobe theclass of positive numbers𝔪 ∈No^>ofthe form𝔪 ={ℝ^>L∣ ℝ^>R} for certain subsets L and R of No with ℝ^>L< ℝ^>R. Numbers in Mo are called monomials. It turns out[11,Theorem 21]that themonomialsforma subgroupof(No^>, ×, <)and that there is a natural isomorphism betweenNoand the orderedfieldℝ[[Mo]]. We will identify those two fields and thus seeNoas afield ofwell-based series. The ordinalω, seen as a surreal number, is the simplest element, or⊑-minimum, oftheclassNo^>,≻.

3 Surreal substructures 3.1 Surreal substructures

In[4], we introduced the notion of surreal substructures. Asurreal substructure is a subclass𝐒 ofNosuch that(No, ⩽,⊑)and(𝐒, ⩽,⊑)are isomorphic. The isomorphismNo⟶ 𝐒isunique and denotedbyΞ_𝐒. Many important subclasses ofNothat are relevant to the study ofhyperserial properties ofNoare surreal substructures. Inparticular, it isknown that thefollowing classes are surreal substructures:

• TheclassesNo^>,No^>,≻andNo^≺of positive,positive infinite and infinitesimal numbers.

• TheclassesMoandMo^≻of monomials and infinitemonomials.

• TheclassesNo≻andNo_≻^>of purely infinite andpositivepurely infinite numbers.

• TheclassMoωoflog-atomicnumbers.

If 𝐔, 𝐕 are surreal substructures, then the class 𝐔^{𝐕 ≔Ξ}𝐔𝐕 is a surreal substructure withΞ_𝐔𝐕=Ξ_𝐔∘Ξ_𝐕.

3.2 Cuts

Given a subclass𝐗ofNoanda∈X, we will write

a^𝐗_L ≔{b∈X:b<a∧b⊑a} and a^𝐗_R ≔{b∈X:b>a∧b⊑a}, so thataL≔a^No_L andaR≔a_R^No. We also writea_⊏^𝐗≔a^𝐗_L∪a^𝐗_R anda_⊏≔a_⊏^No.

IfXis a subclass ofNoandL,Rare subsets ofXwithL<S, then theclass (L∣R)X ≔ {a∈X:(∀l∈L,l<a)∧(∀r∈R,a<r)}

iscalled acut inX. If(L∣R)_Xcontains aunique simplest element, then we denote this element by{L∣R}_Xand say that(L,R)is acut representation(of{L∣R}_X)inX. These notations naturally extend to thecase when𝐋and𝐑are subclasses ofXwith𝐋 < 𝐑.

Asurreal substructure𝐒maybecharacterized as a subclass ofNosuch thatfor allcut repre- sentations(L,R)inS, thecut(L∣R)_Shas aunique simplest element[4,Proposition4.7].

Let S be a surreal substructure. Note that we have a=

{

^a𝐒L∣ a^𝐒_R

}

^{for all a∈S. Let a∈Sand}

let (L,R) be a cut representation of a in S. Then (L,R) is cofinal with respect to (a_L^S,a_R^S) in the sense that L has no strict upperbound ina_L^S and R has no strict lowerbound in a^S_R [4, Proposition4.11(b)].

Given numbersa,b∈Nowitha⩽b, the numberc≔{a_L∣bR}is theunique⊑-maximal number withc⊑a,b. We havea⩽c⩽b. LetSbe a surreal substructure. Consideringthe isomorphism ΞS:(No, ⩽, ⊑)⟶(S, ⩽, ⊑), we see that for alla,b∈S witha⩽b, there is aunique⊑-maximal elementc ofSwithc⊑a,b, and we havea⩽c⩽b. In whatfollows, we willuse thisbasic fact several times withoutfurthermention.

3.3 Cutequations

Let X⊆No be a subclass, let𝐓be a surreal substructure andF:X⟶ 𝐓be afunction. Let λ,ρ befunctions definedforcut representations inXand such thatλ(L,R),ρ(L,R)are subsets of𝐓 whenever(L,R)is acut representation inX. We say that(λ,ρ)is acut equation forF if for all a∈X, we have

λ(a_L^X,a_R^X)<ρ(a^X_L,a^X_R), F(a)={λ(a^X_L,a^X_R)∣ρ(a^X_L,a^X_R)}_𝐓.

Elements inλ(a^X_L,a_R^X) (resp. ρ(a_L^X,a^X_R))arecalledleſt (resp. right)optionsofthiscut equation ata.

We say that thecut equation isuniformifwe have

λ(L,R)<ρ(L,R), F({L∣R}X)={λ(L,R)∣ρ(L,R)}_𝐓

whenever(L,R)is acut representation inX. For instance,givenr∈ ℝ,consider the translation Tr:No⟶No;a⟼a+ronNo. By[19,Theorem 3.2], we have thefollowing uniform cut equa- tionforTronNo:

∀a∈No, a+r={a_L+r,a+r_L∣a+r_R,a_R+r}. (3.1) We will need thefollowingresultfrom [4]:

Proposition 3.1. [4,Proposition4.36]Let 𝐒, 𝐓be surreal substructures. Let Λbe a function from 𝐒to the class of subsets of 𝐓such that for x,y∈ 𝐒with x<y, the set Λ(y)is cofinal with respect toΛ(x). For x∈ 𝐒, let Λ[x]denote the class of elements u of 𝐒such thatΛ(x)andΛ(u)are mutually cofinal. Let {λ∣ρ}_𝐓be a cut equation on𝐒that is extensive in the sense that

∀x,y∈ 𝐒,

(

x⊑y⟹

(

^λ(xL𝐒,x_R^𝐒)⊆λ

(

y_L^𝐒,y_R^𝐒

)

^∧ρ(xL𝐒,x_R^𝐒)⊆ρ

(

y_L^𝐒,y_R^𝐒

)))

^.

Let F: 𝐒 ⟶ 𝐓be strictly increasing with cut equation

∀x∈ 𝐒, F(x)={Λ(x),λ(x_L^S,x_R^S)∣ρ(x_L^S,x_R^S)}_𝐓.

Then F induces an embedding (Λ[x], ⩽, ⊑)⟶(𝐓, ⩽, ⊑)for each element x of 𝐒.

3.4 Convexpartitions

One natural way to obtain surreal substructures isviaconvex partitions. If𝐒is a surreal sub- structure, then aconvex partitionof𝐒is apartitionΠof𝐒whosemembers areconvexsubclasses of𝐒for the order⩽. Wemay thenconsider theclassSmp_Πofsimplest elements(i.e.⊑-minima) in eachmember of Π. Those elements are said Π-simple. For a∈ 𝐒, we letΠ[a]denote the uniquemember ofΠcontaininga. By[4,Proposition4.16], theclassΠ[a] contains aunique Π-simple element, which we denotebyπΠ(a). ThefunctionπΠis a surjective non-decreasing function𝐒 ⟶Smp_ΠwithπΠ∘πΠ=πΠ.

Givena,b∈Smp_Π, note that we havea<bifand only ifΠ[a]<Π[b]. For𝐗 ⊆No, we write Π[𝐗]= ⋃_a∈𝐗Π[a]. We have thefollowing criterion tocharacterize elements ofSmp_Π.

Proposition 3.2. [4, Lemma6.5]An element a of 𝐒is Π-simple if and only if there is a cut representation (L,R)of a in𝐒with Π[L]<a<Π[R]. Equivalently a∈ 𝐒isΠ-simple if and only if Π

[

a_L^𝐒

]

<a<Π

[

a_R^𝐒

]

.

We say thatΠisthinifeachmember ofΠhas acofinal andcoinitial subset. We then have:

Proposition 3.3. [4,Theorem 6.7 andProposition6.8]If Πis thin, then the class Smp_Π is a surreal substructure and ΞSmp_Πhas the following uniform cut equation:

∀z∈No, Ξ_SmpΠz={Π[Ξ_SmpΠzL]∣Π[Ξ_SmpΠz_R]}_𝐒. 3.5 Function groups

Aspecial type ofthinconvex partitions is that of partitions inducedbyfunctiongroups acting on surreal substructures. Afunction group𝒢on a surreal substructure𝐒is a set-sizedgroupof strictly increasing bijections𝐒 ⟶ 𝐒underfunctionalcomposition. We see elementsf,gof𝒢as actions on𝐒and we sometimes writef g andfainstead of f ∘gandf(a), wherea∈ 𝐒.

For such afunctiongroup𝒢, thecollectionΠ_𝒢of classes 𝒢[a] ≔ {b∈ 𝐒 : ∃f,g∈ 𝒢,fa⩽b⩽ga}

witha∈ 𝐒is a thinconvex partition of𝐒. We writeSmp_𝒢≔Smp_Π𝒢. We have theuniform cut equation

∀z∈No, ΞSmp_𝒢z={𝒢ΞSmp_𝒢zL∣ 𝒢ΞSmp_𝒢zR}_𝐒. (3.2) Consider setsX,Y ofstrictly increasing bijections𝐒 ⟶𝐒, then we say thatY ispointwise cofinal with respect toX, and we writeX∠−Y, ifwe have∀f ∈X, ∀a∈ 𝐒, ∃g∈Y,fa⩽ga. We also define

//

^X

//

^{≔ {f}0∘f₁∘ ⋅ ⋅ ⋅ ∘f_n:n∈ ℕ,f₀, . . . ,f_n∈X∪X⁻¹}.

It is easy to see that

//

^X

//

^{is a function group on 𝐒} and that we have

//

^X

//

^∠₋

//

^Y

//

^{if X ∠}₋^{Y or}

X⁻¹∠−Y⁻¹. The relation

//

^X

//

^∠₋

//

^Y

//

trivially implies Smp//^Y//^⊆Smp//^X//^{. If X ∠}−Y and Y ∠−X, then we say that X and Y aremutually pointwise cofinal and we writeX∠∠Y. We then have Smp//^X//^=Smp//^Y//^.

We writeX⩽Y (resp.X<Y)ifwe have∀a∈ 𝐒, ∀f ∈X, ∀g∈Y,fa⩽ga(resp.∀a∈ 𝐒, ∀f ∈X,

∀g∈Y,fa<ga). We also writef <Y andX<ginstead of{f}<Y andX<{g}.

Given afunctiongroup𝒢 on𝐒, the relation definedby f <g⟺{f}<{g}is apartial order on𝒢. We willfrequently rely on thebasic fact that(𝒢,<)ispartially bi-orderedin the sense that

∀f,g,h∈ 𝒢, id_𝐒<g⟺fh<fgh.

3.6 Remarkablefunction groups

Each ofthe examples ofsurreal substructuresfromSubsection3.1canbe regarded as theclasses Smp_𝒢for actions ofthefollowing functiongroups𝒢actingonNo,No^>orNo^>,≻. Forc∈ ℝ andr∈ ℝ^>, we define

Tr ≔ a⟼a+c actingonNoorNo^>,≻. Hc ≔ a⟼r a actingonNo^>orNo^>,≻.

Pc ≔ a⟼a^r actingonNo^>orNo^>,≻. Nowconsider

𝒯 ≔ {T_c:c∈ ℝ}, ℋ ≔ {H_r:r∈ ℝ^>},

𝒫 ≔ {Pr:r∈ ℝ^>},

ℰ′ ≔

//

^{EnHrLn:n∈ ℕ,r∈ ℝ>}

//

^{, and}

ℰ^∗ ≔ {En,Ln:n∈ ℕ}.

Then we have thefollowinglist of correspondences𝒢 ⟼Smp_𝒢:

• The action of𝒯onNo(resp.No^>,≻)yieldsNo≻(resp.No_≻^>), e.g. Smp_𝒯=No≻.

• The action ofℋonNo^>(resp.No^>,≻)yieldsMo(resp.Mo^≻).

• The action of𝒫onNo^>,≻yieldsMo^Mo=E1Mo^≻.

• The action ofℰ′onNo^>,≻yieldsMoω.

• The action ofℰ^∗onNo^>,≻yields𝐊 ≔Moω^No≻(which willcoincide withEωNo_≻^>).

Generalizations ofthosefunctiongroups will allow us to definecertain surreal substructures related to the hyperlogarithms and hyperexponentials onNo.

4 Hyperserialfields

In this section, webriefly recall the definition ofhyperserialfieldsfrom [5]and how toconstruct suchfieldsfromtheir hyperserial skeletons.

4.1 Logarithmichyperseries

Letx be aformal, infinitely large indeterminate. Thefield𝕃of logarithmic hyperseriesof [14]

is the smallest field of well-based series thatcontains all ordinal real power products of the hyperlogarithms Lαx withα∈On. It is naturally equipped with a derivation ∂: 𝕃 ⟶ 𝕃 and composition law∘: 𝕃 × 𝕃^>,≻⟶ 𝕃.

Definition Let α be an ordinal. For each γ<α, we introduce the formal hyperlogarithm

`γ≔L<sub>γ</sub>xand define𝔏<sub><α</sub>tobe thegroupof formalpowerproducts𝔩= ∏<sub>γ<α</sub>`_γ^𝔩γwith𝔩_γ∈ℝ. This group comes with amonomial ordering≻that is definedby

𝔩 ≻1⟺ 𝔩_β>0 forβ=min{γ<α: 𝔩γ≠0}.

We define𝕃<αtobe the orderedfield ofwell-based series𝕃<α≔ ℝ[[𝔏<α]].If α,βare ordinals withβ<α, then we define𝔏_[β,α)tobe the subgroupof𝔏<αof monomials𝔩with𝔩γ=0 whenever γ<β. As in[14], we write

𝕃_[β,α) ≔ ℝ[[𝔏_[β,α)]]

𝔏 ≔

α∈On

𝔏_<α

𝕃 ≔ ℝ[[𝔏]].

We have natural inclusions𝔏_[β,α)⊆ 𝔏<α⊂ 𝔏, hence natural inclusions𝕃_[β,α)⊆ 𝕃<α⊂ 𝕃.

Derivation on𝕃<α Thefield𝕃<αis equipped with a derivation∂: 𝕃<α⟶ 𝕃<αwhich satis- fies the Leibnizrule and which is strongly linear. Write`<sub>γ</sub><sup>†</sup>≔ ∏<sub>ι⩽γ</sub>`ι−1∈ 𝔏_<αfor allγ<α. The derivative ofa logarithmichypermonomial𝔩 ∈ 𝔏<αis definedby

∂𝔩 ≔

(((((((((((

^{γ<α 𝔩γ`γ†}

)))))))))))

^𝔩.

So∂`γ= ¹

∏_ι<γ`ιfor allγ<α. For f ∈ 𝕃_<αandk∈ ℕ, we will sometimes writef^(k)≔ ∂^kf. Composition on 𝕃<α Assume that α=ω^ν for a certain ordinal ν. Then the field 𝕃<α is equipped with acomposition∘: 𝕃<α× 𝕃^>,≻_<α ⟶ 𝕃<α that satisfies inparticular:

• For g∈ 𝕃^>,≻_<α, the map 𝕃<α⟶ 𝕃<α;f ⟼f ∘g is a strongly linear embedding [14, Lemma6.6].

• For f ∈ 𝕃_<αandg,h∈ 𝕃^>,≻_<α, we haveg∘h∈ 𝕃^>,≻_<α andf ∘(g∘h)=(f ∘g)∘h[14,Propo- sition7.14].

• Forg∈ 𝕃^>,≻_<α and successor ordinalsμ<ν, we have`<sub>ω</sub><sup>μ</sup>∘`_ω^μ−=`_ω^μ−1 [14, Lemma5.6].

The sameproperties holdfor thecomposition∘: 𝕃 ×𝕃^>,≻⟶𝕃if αis replacedbyOn.Forγ<α, themap𝕃_<α⟶ 𝕃_<α;f ⟼f ∘`<sub>γ</sub>is injective, with image𝕃<sub>[γ,α)</sub>[14, Lemma5.11]. Forg∈ 𝕃<sub>[γ,α)</sub>, we defineg<sup>↑γ</sup> tobe theunique series in𝕃<αwithg<sup>↑γ</sup>∘`γ=g.

4.2 Hyperserialfields

Let𝔐be an orderedgroup. Areal powering operationon𝔐is a law ℝ × 𝔐 ⟶ 𝔐;(r, 𝔪)⟼ 𝔪^r

oforderedℝ-vector space on𝔐.Let𝕋 = ℝ[[𝔐]] be afield ofwell-based series with𝔐 ≠1, let ν⩽On, and let∘: 𝕃 ×𝕋^>,≻⟶ 𝕋be afunction.Forμ⩽ν, we define𝔐_ω^μtobe theclass ofseries s∈ 𝕋^>,≻with∀γ<ω^μ,`γ∘s∈ 𝔐^≻. We say that(𝕋, ∘)is ahyperserial field if

HF1. 𝕃 ⟶ 𝕋;f ⟼f ∘sis a strongly linearmorphismofordered ringsfor eachs∈ 𝕋^>,≻.

HF2. f ∘(g∘s)=(f ∘g)∘sfor allf ∈ 𝕃,g∈ 𝕃^>,≻, ands∈ 𝕋^>,≻.

HF3. f ∘(t+δ)= ∑_k∈ℕ^f(k)_k!^∘tδ^kfor allf ∈ 𝕃,t∈ 𝕋^>,≻, andδ∈ 𝕋withδ≺t.

HF4. `_ω↑γμ

∘s<`_ω↑γμ

∘tfor all ordinalsμ,γ<ω^μ, ands,t∈ 𝕋^>,≻withs<t.

HF5. Themapℝ^>× 𝔐^≻→ 𝔐;(r,𝔪)↦ 𝔪^r≔`₀^r∘ 𝔪extends to a realpoweringoperation on𝔐.

HF6. `<sub>1</sub>∘(s t)=`₁∘s+`<sub>1</sub>∘t for alls,t∈ 𝕋<sup>>,≻</sup>. HF7. supp`1∘ 𝔪 ≻1 for all𝔪 ∈ 𝔐^≻;

supp`<sub>ω</sub><sup>μ</sup>∘ 𝔞 ≻(`γ∘ 𝔞)⁻¹for all1⩽μ<ν,γ<ω^μand𝔞 ∈ 𝔐_ω^μ.

For eachμ∈On, we define thefunction L_ω^μ: 𝔐_ω^μ⟶ 𝕋; 𝔞 ⟼`_ω^μ∘ 𝔞. The skeleton of (𝕋, ∘)is defined tobe the structure(𝕋,(L_ω^μ)_μ∈On)equipped with the realpower operationfromHF5.

We say that(𝕋, ∘)isconfluentif for allμ∈Onwithμ⩽ν, we have

∀s∈ 𝕋^>,≻, ∃𝔞 ∈ 𝔐_ω^μ, ∃γ<ω^μ, `γ∘s≍`γ∘ 𝔞.

Inparticular(𝕃, ∘)is aconfluent hyperserialfield.

4.3 Hyperserial skeletons

It turns out that each hyperlogarithmL_ω^μon a hyperserialfield𝕋canuniquelybe reconstructed from its restriction to the subset of L_<ω^μ-atomic hyperseries (here we say that f ∈ 𝕋^>,≻ is L_<ω^μ-atomicif Lγf ∈ 𝔐for allγ<ω^μ). One ofthemain ideas behind[14]is to turn thisfact into a way toconstruct hyperserialfields. This leads to the definition ofa hyperserial skeleton as afield𝕋withpartially defined hyperlogarithmsL_ω^μ, which satisfy suitablecounterparts of the above axiomsHF1untilHF7.

Moreprecisely, let𝕋= ℝ[[𝔐]] be afield ofwell-based series andfixν∈On^>∪{On}. Ahyper- serial skeleton on 𝕋 of force ν consists of a family of partial functions L_ω^μ for μ<ν, called (hyper)logarithms, which satisfy a list ofaxioms that we will describe now.

First ofall, the domains𝔐_ω^μ≔domL_ω^μon which thepartialfunctionsL_ω^μare defined should satisfy thefollowingaxioms:

Domains ofdefinition:

DD₀. domL₁= 𝔐^≻;

DDμ. domL_ω^μ= ⋂_η<μdomL_ω^η, if μis a non-zero limit ordinal;

DDμ. domL_ω^μ={s∈ 𝕋:L_ω^∘nμ−(s)∈domL_ω^μ−for alln}, if μ is a successor ordinal.

It willbeconvenient to also define theclass𝔐_ω^νby

𝔐ω^ν ≔ {s∈ 𝕋:L_ω^∘nν−(s)∈ 𝔐ω^ν−for alln} if νis a successor ordinal 𝔐_ω^ν ≔

μ<ν

𝔐_ω^μ if νis a non-zero limit ordinal.

Consider an ordinalγ<ω^νwritten in Cantor normalformγ= ∑_i=1^r ω^ηiniwhereη₁>η₂> ⋅⋅⋅ >ηr

andn₁, . . . ,nr<ω. We denotebyLγ thepartialfunction

Lγ ≔ L_ω∘nη11∘ ⋅ ⋅ ⋅ ∘L_ω∘nηrr. (4.1) Itfollowsfromthe definition thatfor allμ⩽ν, theclass𝔐_ω^μconsists ofthose seriess∈ 𝕋^>,≻

for whichs∈domLγandLγs∈ 𝔐^≻for allγ<ω^μ. Wecall such seriesL_<ω^μ-atomic.

Secondly, the hyperlogarithmsL_ω^μwithμ<ν should satisfy thefollowingaxioms:

Axioms for the logarithm Functional equation:

FE₀. ∀𝔪, 𝔫 ∈ 𝔐₁,L₁(𝔪 𝔫)=L₁𝔪 +L₁𝔫.

Asymptotics:

A₀. ∀r∈ ℝ^>, ∀𝔪 ∈ 𝔐₁,L₁𝔪 ≺ 𝔪.

Monotonicity:

M₀. ∀𝔪, 𝔫 ∈ 𝔐₁, 𝔪 ≺ 𝔫 ⟹L₁𝔪 <L₁𝔫.

Regularity:

R₀. ∀𝔪 ∈ 𝔐₁,suppL₁𝔪 ≻1.

Surjective logarithm:

SL. ∀φ∈ 𝕋_≻^>, ∃𝔪 ∈ 𝔐₁,φ=L₁𝔪.

Axioms for the hyperlogarithms(for eachμ∈Onwith0<μ<νandβ≔ω^μ) Functional equation:

FEμ. ∀𝔞 ∈ 𝔐_β,L_βL_β/ω𝔞 =L_β𝔞 −1if μis a successor ordinal.

Asymptotics:

Aμ. ∀γ<β, ∀𝔞 ∈ 𝔐_β,L_β𝔞 <Lγ𝔞.

Monotonicity:

Mμ. ∀𝔞, 𝔟 ∈ 𝔐_β, ∀γ<β, 𝔞 ≺ 𝔟 ⟹L_β𝔞 +(L_γ𝔞)⁻¹<L_β𝔟 −(L_γ𝔟)⁻¹. Regularity:

Rμ. ∀𝔞 ∈ 𝔐_β, ∀γ<β,suppL_β𝔞 ≻(Lγ𝔞)⁻¹.

Finally,forμ⩽νwithμ∈On, we also need thefollowingaxiom Infiniteproducts:

Pμ. ∀𝔞 ∈ 𝔐_β, ∀𝔩 ∈ 𝔏_<β^≻ , ∑_γ<β𝔩γLγ+1𝔞 ∈L₁𝔐^≻.

Note thatSLandR₀together implyL₁𝔐^≻= 𝕋_≻^>, whencePμautomatically holds. This will in particularbe thecaseforNo(see Section5).

In summary, we have:

Definition4.1.[5,Definition3.3]Givenν∈On^>∪{On}, we say that(𝕋,(L_ω^μ)μ<ν)is ahyper- serial skeletonof forceνif it satisfiesDD_μ,FE_μ,A_μ,M_μ, andR_μ for all μ<ν, as well asP_μ for all ordinals μ⩽ν.

Assume that𝕋is a hyperserial skeleton of forceν. Thepartial logarithmL₁: 𝔐₁⟶ 𝕋extends naturally into a strictly increasing morphism(𝕋^>, ×,<)⟶(𝕋,+, <), which wecall thelogarithm and denotebyL₁orlog [5, Section4.1]. If𝕋satisfiesSL, then this extended logarithmis actu- ally an isomorphism [29,Proposition2.3.8]. In thatcase,for anys∈ 𝕋^>andr∈ ℝ, we define s^r≔exp(rlogs)∈ 𝕋^>.

4.4 Confluence

Definition4.2. [5,Definition3.5]Given a hyperserial skeleton𝕋 = ℝ[[𝔐]]of force ν∈On^>

and μ<ν, we inductively define the notion of μ-confluencein conjunction with the definition of functions𝔡_ω^μ: 𝕋^>,≻⟶ 𝔐_ω^μ, as follows.

• The field 𝕋 is said 0-confluent if 𝔐is non-trivial. The function 𝔡₁ maps every positive infinite series s∈ 𝕋^>,≻onto its dominant monomial𝔡s. For each s∈ 𝕋^>,≻, we write

ℰ₁[s] ≔ {t∈ 𝕋^>,≻:t≍s}.

Let μ⩽νbe such that 𝕋is η-confluent for all η<μ and let s∈ 𝕋^>,≻.

• If μ is a successor ordinal, then we writeℰ_ω^μ[s]for the class of series t with (L_ω^μ−∘ 𝔡_ω^μ−)^∘n(s) ≍ (L_ω^μ−∘ 𝔡_ω^μ−)^∘n(t)

for a certain n∈ ℕ.

• If μ is a limit ordinal, then we writeℰ_ω^μ[s]for the class of series t with L_ω^η𝔡_ω^η(s) ≍ L_ω^η𝔡_ω^η(t)

for a certain η<μ.

We say that 𝕋 is μ-confluent if each class ℰ_ω^μ[s] contains a L_<ω^μ-atomic element; we then define 𝔡_ω^μ(s)to be this element.

This inductive definition is sound. Indeed, if μ⩽ν+1and𝕋isη-confluentfor allη<μ, then thefunctions𝔡_ω^η: 𝕋^>,≻⟶ 𝔐_ω^ηwithη<μare well-defined and non-decreasing. Thus,forη<μ, thecollection ofℰ_ω^η[s]withs∈ 𝕋^>,≻forms apartition of𝕋^>,≻intoconvexsubclasses.

We say that 𝕋 is confluent if it is ν-confluent. If 𝕋 has force On, then we say that 𝕋 is On-confluent, orconfluent, if(𝕋,(L_ω^η)_η<μ)isμ-confluentfor allμ∈On.

4.5 Correspondencebetween fields andskeletons

Proposition 4.3. [5,Theorem 1.1]If (𝕋,(L_ω^μ)μ∈On)is a confluent hyperserial skeleton, then there is a unique function ∘: 𝕃 × 𝕋^>,≻⟶ 𝕋with

∀μ∈On, ∀𝔞 ∈ 𝔐_ω^μ, `_ω^μ∘ 𝔞 =L_ω^μ𝔞 such that(𝕋, ∘)is a confluent hyperserial field.

Assume now that𝕋is only a hyperserial skeleton of force ν∈On^>∪{On}and thatμ is an ordinal with0<μ<ν such that(𝕋,(L_ω^η)_η<μ)isμ-confluent. Letβ≔ω^μ. By[5,Definition4.11 and Lemma4.12], thepartialfunctionL_β naturally extends into afunction𝕋^>,≻⟶ 𝕋^>,≻that we still denotebyL_β. This extendedfunction is strictly increasing,by‘ [5, Corollary4.17]. If μ is a successor ordinal, then it satisfies thefunctional equation

∀s∈ 𝕋^>,≻, L_βL_β/ωs=L_βs−1, (4.2)

by [5,Proposition 4.13]. For γ<β, we have a strictly increasing function L_γ: 𝕋^>,≻⟶ 𝕋^>,≻

obtained as acomposition of functionsL_ω^ηwithη<μ, as in(4.1). By[5,Proposition4.7], we have ℰ_β[s] = {t∈ 𝕋^>,≻: ∃γ<β,Lγt≍Lγs}.

4.6 Hyperexponentiation

In a traditional transseries field 𝕋, the transmonomials are characterizedby thefact that,for any f ∈ 𝕋^>, we have

f ∈ 𝔐 ⟺ supplogf ≻1. (4.3)