Sign sequences of log-atomic numbers

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Vincent Bagayoko

To cite this version:

Vincent Bagayoko. Sign sequences of log-atomic numbers. 2020. �hal-02952455�

Vincent Bagayoko^∗,UMons, LIX September 29, 2020

Log-atomic numbers are surreal numbers whose iterated logarithms are monomials, i.e. additively irreducible num- bers. Presenting surreal numbers as sign sequences, we give the sign sequence formula for log-atomic numbers. In doing so, we relate log-atomic numbers to fixed-points of certain surreal functions.

INTRODUCTION

The class Noof surreal numbers of J. H. Conway [7] is an inductively defined ordered field with additional structure. Conway uses the abstract notion of Dedekind inspired cut as a fun- damental constructor to define numbers by well-founded induction.

Indeed, any numberx∈Nois obtained from setsL,R of previously defined numbers, as the

“simplest” number withL<xandx<R. This relation is denotedx= {L|R}. Conversely, any sets of surreal numbers L,R withL<R give rise to a unique simplest number{L|R}. Thus the def- inition ofNocomes with several features: an inductively defined order<, an order-saturation property, a corresponding ordinal rank called the birthday β(x)which represents the minimal ordinal number of inductive steps required to yield x. For instance0≔ {∅|∅}has birthday0 whereas1≔ {{0}|∅}has birthday1 and¹/₂≔ {{0}|{1}}has birthday2.

The versatility of this construction allowed Conway and several authors aſter him to define a rich structure on No. In particular, he defined ring operations that are compatible with the ordering and turn(No, +, ×, ⩽)into an ordered field extension of the reals, as well as an exten- sion of the ordered semi-ring (On, ⊕, ⊗, ∈) of ordinal numbers under Hessenberg sum and product.

Conway also discovered thatNoenjoys a natural structure of field of Hahn series as per [10].

Every surreal number can be expressed as a possibly transfinite sum of additively irreducible numbers calledmonomials. Moreover, monomials can be parametrized by a morphismx⟼ω˙^x, called the ω-map, for which Conway gave a an equation using the cut presentation of num- bers. Whereas the latter presentation is useful to produce inductive definitions of functions, by induction on the birthday for instance, the presentation of numbers as Hahn series is useful to compute certain operations on No. In particular, it can be used to describe H. Gonshor's exponential function [9], and A. Berarducci and V. Mantova's surreal derivation [5].

The sign sequence presentation of surreal numbers, invented and studied by H. Gonshor in [9], is a way to give precise description of surreal numbers. In this picture, numbers are sequences of signs +1, −1 indexed by ordinals, or equivalently, nodes in the binary tree {−

1, +1}^<On. The birthdate measure then coincides with the domain, calledlength, of the sign sequences. A natural relation ofsimplicity, which corresponds to the inclusion of sign sequences in one another and is denoted⊑, emerges as a more precise measure of the complexity of num- bers.

Sign sequences are by no means an optimal tool to describe surreal numbers in the context of the ordered field, exponential ordered field, Hahn series field, or differential field structures.

For instance given numbersx,y with known sign sequences, computing the sign sequence of x+y, or even formulating the properties of that ofx yin general are open problems. However, sign sequences behave relatively well with respect to operations that preserve simplicity under certain conditions. This includes theω-map, the transfinite summation which identifies num- bers with Hahn series, and the exponentiation of monomials. In this context, sign sequences can be useful tools to understand the behavior of length under the operations, as well as deriving general properties of certain classes of numbers with respect to the relations⩽and⊑.

∗. UMons, LIX, vincent.bagayoko@lilo.org

Describing the sign sequences of elements in a class𝐘oſten requires a reference with respect to which they can be given. One way to give such a reference is to find a parametrization of𝐘, i.e. a bijectionF:X⟶Yfor a certain subclass𝐗 ⊆No, and then describe the sign sequence of f(x) =yin terms of that ofx for eachx∈ 𝐗. This presumes that the behavior of functionF on sign sequences may be understood, suggesting thatF should be compatible in a sense with the relations⩽and⊑. There enters the notion of surreal substructure of [3]. Surreal substructures are subclasses 𝐒ofNo that are isomorphic to(No, ⩽, ⊑)under the restrictions of ⩽and⊑to 𝐒 × 𝐒. The isomorphism Ξ_𝐒:No⟶ 𝐒 then being unique, one may rely on it to relate the sign sequence of Ξ_𝐒z∈ 𝐒with that ofz.

Surreal substructures naturally appear when defining certain operations onNo, see for instance [3, Sections 6 and 7]. The study of surreal substructure yields tools to express and compute sign sequences. In particular, expressing a structure Ξ_𝐒using classes of fixed points for given parametrizations yields a method to compute the sign sequence formula of Ξ_𝐒. Relying exten- sively on [3], we will develop the relevant notions in Sections1and2.

A surreal substructure of particular interest is the classLaof log-atomic numbers. Those are numbers𝔞such that then-fold iteration oflog^∘n𝔞of the logarithm at𝔞yields a monomial for eachn∈ ℕ. This structure plays a crucial role in the definition of derivations onNothat are compatible with the exponential and the structure of field of series ofNo. It is also used in the investigation of the properties of expansions of numbers as transseries. This class was char- acterized by Berarducci and Mantova [5, Section 5] who defined such a derivation and proved a fundamental structure property forNo. FinallyLaplays a role in the definition of the first surreal hyperexponential function Eω[4]. Our our goal in this article is to compute the sign sequence formula for ΞLa. This will in particular give the sign sequence for Eω(a)for each surreal numbera> ℝwhose length is strictly below the firstε-numberε0∈On.

In this task, we are continuing work of S. Kuhlmann and M. Matusinski in [13]. Indeed, they considered a surreal substructure𝐊properly contained inLaand determined its sign sequence formula. The relation between𝐊andLawas subject to the conjecture

La=^?log^∘ℤ𝐊 [13, Conjecture 5.2],

which turned out to be false [5, Proposition 5.24]. The correct relation betweenLaandKwas later found by Mantova and Matusinski [14]. We will rely on their sign sequence formula and a presentation ofLausing classes of fixed points to derive our formula.

1 Numbers and sign sequences 1.1 Numbers as sign sequences

As Gonshor, we define numbers as sign sequences.

D^EFINITION1. A surreal number is a map x: ℓ(x) → {−1,1};α↦x[α], where ℓ(x) ∈Onis an ordinal number. We call ℓ(x)thelengthof x and the map x: ℓ(x) → {−1,1}thesign sequence of x. We writeNofor the class of surreal numbers.

Given a surreal number x∈No, we extend its sign sequence withx[α] =0 for all α⩾ ℓ(x).

Givenx∈Noandα∈On, we also introduce therestriction y=x↿α∈Notoαas being the initial segment of xof lengthα, i.e. y[β] =x[β]forβ<αandy[β] =0 forβ⩾α.

The ordering ⩽ on No is lexicographical: given distinct elements x,y∈No, there exists a smallest ordinalαwithx[α] ≠y[α]and we setx<yif and only if x[α] <y[α].

For x,y∈No, we say thatx issimpler thany, and writex⊑y, if x=y↿ ℓ(x). The partially ordered class(No, ⊑)is well-founded. We writex⊏yifx⊑yandx≠y. Forx∈No, we write

x⊏≔ {a∈No:a⊏x}

for the set of numbers that are strictly simpler thanx. The set(x_⊏, ⊑)is well-ordered with order typeot(x_⊏, ⊑) = ℓ(x). Moreover,x_⊏is the union of the sets

xL≔ {y∈No:y⊏x,y<x} and xR≔ {y∈No:y⊏x,y>x}.

Every linearly ordered subsetX of(No, ⊑)has asupremum s=sup_⊑X inNo. Indeed, we have ℓ(s) =supℓ(Z), ands[α] =x[α]for allx∈X withα< ℓ(x). Numbersxthat are equal tosup_⊑x_⊏ are calledlimit numbers; other numbers are calledsuccessor numbers. Limit numbers are exactly the numbers whose length is a limit ordinal.

Notice thatNois a proper class. For instance, the linearly ordered class(On, ⩽)is embedded into the partial order(No, ⊑)through the mapα⟼ (1)_β<α:On⟶No. We will thus identify ordinals numbers as surreal numbers.

1.2 Monomials

Any non-archimedean ordered field Κ admits a non-trivial valuation called the natural valuation [1, Section 3.5]. The natural valuation ofx∈Κ^>is related to itsarchimedean class, i.e. the class

ℋ[x] ≔ {y∈Κ^>: ∃q∈ ℚ^>,q⁻¹x<y<q x}.

In NBG, the corresponding valued ordered field can be embedded [11] into a field of generalized Hahn series, as defined by Hahn [10]. The existence of such embeddings, called Kaplansky embeddings, usually requires the axiom of choice.

Having defined the structure of ordered field, Conway noticed that for No, the instances where choice is required could be circumvented by the use of the inductive nature ofNo. Indeed, inNo, each archimedean classℋ[x]has a unique⊑-minimal element denoted𝔡x, and the class Mo≔ {𝔡x:x∈No^>} is a subgroup of (No^>, ×) which can be used to carry the definition of the Kaplansky embedding.

D^EFINITION2. Amonomialis a number 𝔪which is simplest in the classℋ[𝔪], i.e. a number of the form𝔪 = 𝔡xfor a certain x∈No^>.

Moreover, Conway defined a parametrization z⟼ω˙^z of Mo. This is a strictly increasing morphism(No, +, ⩽) ⟶ (Mo, ×, ⩽)which also preserves simplicity:

∀x,y∈No,x⊑y⟺ω˙^x⊑ω˙^y.

The operationα⟼ω˙^α:On⟶Nocoincides with the ordinal exponentiation with basisω.

1.3 Arithmetic on sign sequences

For ordinalsα,β, we will denote their ordinal sum, product, and exponentiation byα∔β,α×.β andα˙^β. Here, we introduce the operations∔and×. of [3, Section 3.2] onNo. Those are natural extensions of ordinal arithmetic toNo, which we will use in order to describe sign sequences.

For numbersx,y, we writex∔yfor the number whose sign sequence is the concatenation of that of yat the end of that of x. Sox∔y is the number of lengthℓ(x∔y) = ℓ(x) ∔ ℓ(y), which satisfies

(x∔y)[α] = x[α] (α< ℓ(x))

(x∔y)[ℓ(x) ∔β] = y[β] (β< ℓ(y))

The operation∔clearly extends ordinal sum. We havex∔0=0∔x=xfor allx∈No.

We writex×.yfor the number of lengthℓ(x) ×.ℓ(y)whose sign sequence is defined by (x×.y)[ℓ(x) ×.α∔β] = y[α]x[β] (α< ℓ(y),β< ℓ(x))

The operation×. extends ordinal product. Special cases can be described informally: givenx∈ No and α∈On, the numberx×. α is the concatenation of x with itself to the right “α-many times”, whereas α×. x is the number obtained fromx by replacing each sign by itself α-many times. We havex×.1=x andx×.(−1) = −xfor allx∈No.

The operations also enjoy the properties which extend that of their ordinal counterparts as is illustrated hereaſter. We refer to [3, Section 3.2] for more details.

LEMMA3. [3, Lemma 3.1]For x,y,z∈No, we have

a) x∔ (y∔z) = (x∔y) ∔z, x∔0=0∔x=x, and x∔y=sup_⊑(x∔y⊏)if y is a limit.

b) x×.(y×.z) = (x×.y) ×.z, x×.(y∔z) = (x×.y) ∔ (x×.z),and x×.y=sup_⊑(x×.y_⊏)if y is a limit.

1.4 Sign sequences formulas

Ifαis an ordinal andf is a functionα⟶No, then(α,f)uniquely determines a surreal number [α,f]defined inductively by the rules

• [0,f] =0,

• [γ∔1,f ↿ (γ+1)] = [γ,f ↿γ] ∔f(γ)forβ<α,

• [γ,f ↿γ] =sup_⊑{[μ,f ↿μ] :μ<γ}for limit ordinalsγ⩽α.

We say that (α,f)is a sign sequence formula for[α,f], which we identify with the informal expression

˙

γ<αf(γ) =f(0) ∔f(1) ∔ ⋯ ∔f(β) ∔ ⋯

whereγranges inα. For instance, the ordered pair(ω, ((−1)ⁿ)n<ω)is a sign sequence formula of¹/₂×.ω=1∔ (−1)∔1∔ (−1)∔ ⋯ = /²₃. In general, we look for such formulas wheref alternates betweenOn^>and−On^>, that is, where f ranges inOn^>∪ −On^>and where, for each ordinal β withβ∔1<α, we havef(β)f(β∔1) <0. It is easy to see that every surreal number admits a unique such alternating sign sequence formula. We refer to this formula asthe sign sequence formula of said number.

If F:No⟶Nois a function, we may look for a function Ψ whose value at each numberxis a sign sequence formula of F(x). We then consider Ψ as asign sequence formulaforF.

As an example, we now state Gonshor's results regarding the sign sequences of monomials.

For z∈No, we writeτzfor the order type of(zL, ⊑), that is the order typez⁻¹[{1}]whenz is seen as a sign sequence, or equivalently the ordinal number of signs1 in the sign sequence ofz. We haveτz=0 if and only ifz∈ −On. Gonshor found the following sign sequence formula for monomials:

PROPOSITION4. [9, Theorem 5.11]Let z be a number. The sign sequence formula for ω˙^zis ω˙^z=1∔ ˙

β<ℓ(x) z[β]ω˙^{τz↿(β+1)+1} . For instance, we have

ω

√ = ω˙^1/2=ω∔ (−ω²) and logω = ω˙^ω−1=ω∔ (−ω³).

Let us also specify a consequence that we will use oſten in what follows.

L^EMMA5.Let x,y∈Nobe such that the maximal ordinal α with α⊑y is a limit. We have ω˙^x∔y= ω˙^x∔ω˙^τx∔y.

PROOF. Letαbe maximal withα⊑yand writey=α∔z. Sinceyis strictly positive, we haveα>

0, whenceα⩾ω. Let

𝔪 ≔ z[0]ω˙^{ττx∔α∔z↿1+1} ∔ z[1]ω˙^{ττx∔α∔z+1} ∔ ⋯ ∔ z[γ]ω˙^{ττx∔α∔(z↿(γ+1))+1} ∔ ⋯

whereγranges inℓ(z). Proposition4yieldsω˙^τx∔y=ω^τx∔α∔z=ω^τx∔α∔ 𝔪. On the other hand, we haveω^x∔y=ω^x∔ 𝔫 ∔ 𝔪where𝔫is defined as the transfinite concatenation

𝔫 = (y[0]ω˙^τx∔1+1) ∔ ⋯ ∔ y[β]ω^τx∔β+1 ∔ ⋯, whereβranges inα. Forβ<α, we havey[β] =1 andτ_x∔β=τ_x∔β, so

𝔫 = ω˙^τx∔1+1∔ ⋯ ∔ω˙^τx∔β+1∔ ⋯

= ω˙^τx∔α.

It follows thatω˙^x∔y=ω˙^x∔ω˙^τx∔y. □

2 Surreal substructures 2.1 Surreal substructures

D^EFINITION6. Asurreal substructureis a subclass𝐒of Nosuch that(No,⩽, ⊑)and(𝐒, ⩽,⊑) are isomorphic. The isomorphism (No, ⩽, ⊑) ⟶ (S, ⊑, ⩽) is unique, denoted Ξ_𝐒and called the parametrizationof 𝐒.

Example7. Here are examples of common surreal substructures and corresponding parame- trizations.

• Forx∈No, the classx∔No=No^⊒x≔ {y∈No:x⊑y}has parametrizationz⟼x∔z. We have

No^⊒1 = No^>≔ {y∈No:y>0}, No^⊒ω = No^>,≻≔ {y∈No:y> ℝ}, No^⊒ω−1 = No^>,≺≔ {y∈No:0<y< ℝ^>}.

• Forx∈No^>, the classx×.No≔ {x×.z:z∈No}has parametrizationz⟼x×.z. We have 1×.No = No

ω×.No = No_≻≔ {y∈No:suppy≻1}.

• The classMoof monomials has parametrizationz⟼ω˙^z. 2.2 Convexity

Let𝐒be a surreal substructure and let𝐂be aconvexsubclass of𝐒. This means that forx,y,z∈𝐒 withx⩽y⩽z, we havex,y∈ 𝐂 ⟹y∈ 𝐂. Then we have a simple criterion to decide whether𝐂 is a surreal substructure:

P^ROPOSITION8. [3, Proposition 4.29]A convex subclass𝐂of 𝐒is a surreal substructure if and only if every subset of 𝐂has strict upper and lower bounds in𝐂.

In particular, non-empty open intervals (a,b) ∩ 𝐒, where a,b∈ 𝐒 ∪ {−∞, +∞}, are surreal substructures. The following proposition gives a sign sequence formula for Ξ_(a,b) in certain particular cases.

LEMMA9. Let α be a non-zero ordinal. Let a,b and z be numbers.

a) We haveΞ_(a,a∔α)z=a∔1∔z if z≯αL,

andΞ_(a,a∔α)(α∔z) =a∔α∔ (−1) ∔z.

b) We haveΞ_{(b∔(−α),b)}z=b∔ (−1) ∔z if z≮ −(α_L), andΞ_{(b∔(−α),b)}((−α) ∔z) =b∔ (−α) ∔1∔z.

PROOF. First notice that we have α⊑z if and only if z> (α_L)and (−α) ⊑z if and only if z< −(α_L)so the given descriptions cover all cases. We will only derive the formulas for the interval(a,a∔α), the other ones being symmetric. The numbera∔1 is the root of(a,a∔α)so forz∈No, we have Ξ_(a,a∔α)z=a∔1∔uzfor a certain numberuz. The classNo^≯αLis convex in Noand forz∈No^≯αL, we havea∔1∔z∈ (a,a∔α). The functionz⟼a∔1∔z defined onNo defines a surreal isomorphism, so we have Ξ_(a,a∔α)z=a∔1∔zifz∈No^≯αL. On the other hand, we have

No∖No^≯αL = No^⊒α, and (a,a∔α) ∖ (a∔1∔No^≯αL) = a∔α∔ (−1) ∔No.

It follows that Ξ_(a,a∔α)↿No^⊒α=Ξ_NoNo⊒a∔α∔(−1)^⊒α , which yields the other part of the description. □ 2.3 Imbrication

LetU,Vbe two surreal substructures. Then there is a unique(⩽, ⊑)-isomorphism Ξ_V^𝐔≔Ξ_VΞ_𝐔⁻¹: 𝐔 ⟶V

that we call the surreal isomorphism between S and T. The composition ΞU∘ΞV is also an embedding, so its imageU^V≔ΞUΞVNois again a surreal substructure that we call theimbri- cation ofVintoU. Givenn∈ ℕ, we write𝐔ⁿfor then-fold imbrication of𝐔into itself. We have Ξ_Un=Ξ_Uⁿ.

Example10. Forx,y∈No, it follows from Lemma3that we have (x∔No)^{(y∔No) = (x∔y) ∔No,}

(x×.No)^{(y×.No) = (x×.y) ×.No, and} (x×.No)^{(y∔No) = (x×.y) ∔ (x×.No).}

Less elementary results include the relationMo^No>=No≻^Moof [3, Proposition 7.3].

The relation of imbrication is related to the inclusion of surreal substructures in the following way:

PROPOSITION11. [3, Proposition 4.11]Given two surreal substructuresU, 𝐕, we have𝐔 ⊆ 𝐕if and only if there is a surreal substructure𝐖with𝐔 = 𝐕^𝐖.

2.4 Fixed points

D^EFINITION12. Let 𝐒be a surreal substructure. We say that a number z is𝐒-fixedif Ξ_𝐒z=z.

We writeS^ωfor the class of 𝐒-fixed numbers.

If𝐔, 𝐕, 𝐖are surreal substructures with𝐔 = 𝐕𝐖, then the classW^ωis that of numbersz with Ξ_𝐔z=Ξ_𝐕z.

P^ROPOSITION13. [3, Proposition 5.2]If 𝐒is a surreal substructure, thenS^ω= ⋂_n∈ℕ𝐒ⁿ. Notice that𝐒may not itself be a surreal substructure in general. We thus consider the fol- lowing notion of closure:

DEFINITION14. We say that𝐒isclosedif we haveΞ_𝐒sup_⊑Z=sup_⊑Ξ_𝐒Z for all non-empty set Z⊂Nosuch that(Z, ⊑)is linearly ordered.

LEMMA15. [3, Lemma 5.12]If Uand𝐕are closed surreal substructures, then so is𝐔^𝐕.

P^ROPOSITION16. [3, Corollary 5.14]If 𝐒is closed, then𝐒^ωis a closed surreal substructure.

In general, when𝐒^ωis a surreal substructure, we write Ξ_𝐒ω ≔ Ξ_𝐒^ω.

Example17. Here are some examples of structures of fixed points (see [3, Example 5.3]).

• Forx∈No, the structureNo^⊒xis closed, with(No^⊒x)^ω=No^⊒x×.ω.

• For x∈No^>, the structure x×. No is closed, with (x×. No)^ω=x^∞×. No where x^∞≔ sup_⊑{x,x×.x,x×.x×.x}.

• The structureMois closed, and the parametrization Ξ_Mo^ω of(Mo)^ωis usually denoted z⟼εz, since it extends the ordinal functionζ⟼ε_ζ which parametrizes the fixed points of ζ⟼ω^ζ.

• The structure 1+Mo^≺(whereMo^≺=Mo∩No^≺) is closed. Moreover, its structure of fixed points (1+Mo^≺)^ωcoincides with the class of fixed points of the functionz⟼ 1+ω˙^z−3/2, although this last one is not a surreal isomorphism. Informal expansions

x=1+ω˙^{−1/2+ω˙−1/2+ω˙}

⋰

,

of numbers in this class can be replaced by the formulation∃z∈No,x=Ξ_1+Mo^{ω ≺}z. The sign sequence ofxcan be computed in terms of that ofzusing Proposition19below.

2.5 Sign sequence formulas for closed structures

Let 𝐒 be a closed surreal substructure and let Ψ_𝐒= (αz,fz)_z∈No denote its alternating sign sequence formula. This formula is “continuous” or “closed” in the sense that for any limit number z, we haveαz=sup_⊑αz⊏ and fz= ⋃fz⊏. If for each surreal numberu and forσ∈ {−

1,1}we writeϕ_u∔σ for the non-zero surreal number defined by(Ξ_𝐒u) ∔ϕ_u∔σ^𝐒 =Ξ_𝐒(u∔σ), we then have

∀z∈No,Ξ_𝐒z=Ξ_𝐒0∔ϕ_z↿1^𝐒 ∔ϕ_z↿2^𝐒 ∔ ⋯ ∔ϕ_z↿(β^𝐒 ₊₁₎∔ ⋯

where β ranges in ℓ(z). Thus it is enough to compute the numbers ϕ_u∔σ to determine sign sequences of element of 𝐒. In the case in Gonshor's formulas for the ω-map and theε-map and Kuhlmann-Matusinski's formula for theκ-map, those numbers are ordinals or opposites of ordinals. If𝐒is not closed, then for every limit numberafor which there will be an additional termι(a) ∈Nosuch thatδ(a)is not always zero, and that we have, for allz∈No,

Ξ_𝐒z=Ξ_𝐒0∔ϕ_z↿1^𝐒 ∔ϕ_z↿2^𝐒 ∔ ⋯ ∔δ(z↿ω) ∔ϕ_z↿(ω+1)^𝐒 ∔ ⋯ ∔δ(z↿ (ω×.η)) ∔⋯ ∔ϕ_z↿(β+1)∔ ⋯, whereω×.η⩽β< ℓ(z). Such termsδ(a)must then be computed independently. Thus it may be a good first step towards computing a sign sequence formula for𝐒to check whether it is closed or not, and where closure defects occur.

Example18. The parametersϕ_z∔σfor theω-map are ϕ_z∔(−1)^Mo = ω˙^τz+1 and

ϕ_z∔1^Mo = ω˙^τz+2 for allz∈No.

By [9, Chapter 9], the sign sequence formula forε⋅is given by the parameters:

ϕ_z∔(−1)^Moω = −ε˙_τ^ω_z and

ϕ_z∔1^Moω = ετz+1 for allz∈No.

Assume again that𝐒is a closed surreal substructure and letz∈No. Our goal in this paragraph is to compute Ξ_S^ωzprovided we know Ξ_S^ωonz_⊏. The identityS^ω= ⋂_n∈ℕΞ_𝐒ⁿNosuggests that Ξ_S^ωmay be a form of limit of Ξ_𝐒ⁿasntends to infinity. Indeed, we claim that one can always find az∈Nosuch that Ξ_𝐒^ℕazis a⊑-chain with supremum Ξ_S^ωz. More precisely, we have:

PROPOSITION19. Let 𝐒is a closed surreal substructure and let z∈No. We have Ξ_S^ωz = sup

⊑

Ξ_𝐒^ωz_⊏ if z is a limit, and Ξ_S^ωz = sup

⊑ Ξ_𝐒^ℕ(Ξ_S^ωu∔σ) of z=u∔σ for certain u∈No,σ∈ {−1,1}.

PROOF. LetZ=Ξ_S^ωz_⊏, and setsz=sup_⊑Z. Since𝐒is closed, so isS^ω, which implies thatszis 𝐒-fixed.

Assume that zis a limit number. Notice thatszis the simplest element of𝐒^ωwith Ξ_S^ωzL<

sz<Ξ_S^ωzR, whereaszis the simplest number withzL<z<zR. We deduce thatsz=Ξ_𝐒^ωz.

Otherwise, there is a numberuand a signσ∈ {−1,1}withz=u∔σ. Let az≔ (Ξ_S^ωu) ∔σ.

Since the sign of az− (Ξ_S^ωu)isσ, the sign of Ξ_𝐒az−Ξ_𝐒(Ξ_S^ωu) =Ξ_𝐒az− (Ξ_S^ωu)is σ as well, soaz⊑Ξ_𝐒az. It follows by induction that Ξ_𝐒^ℕazis a⊑-chain whose supremumy lies in each Ξ_𝐒ⁿSforn∈ ℕby closure of those structures, soy∈S^ω. Since the sign of Ξ_S^ωz−Ξ_S^ωuisσ, we havea_z⊑Ξ_S^ωz, whencey⊑Ξ_S^ωz. Now Ξ_S^ωzis the simplest𝐒-fixed number such that the sign of

Ξ_S^ωz−Ξ_S^ωuisσ, whencey=Ξ_S^ωz. □

3 Exponentiation

In this section, we define the exponential function of [9, Chapter 10] as well as the surreal substructuresLaand𝐊.

3.1 Surreal exponentiation

Inductive equations Recall that by Conway's construction, or in Gonshor's setting by [9, Theorem 2.1], given setsL,Rof numbers withL<R, there is a unique⊑-minimal, or simplest, number{L|R}with

L< {L|R} <R.

Let𝐒be a surreal substructure. Carrying the previous property through the parametrization of 𝐒, we obtain, given a surreal substructure𝐒and subsetsL,R⊂ 𝐒withL<R, a unique⊑-minimal element{L|R}_𝐒of𝐒with

L< {L|R}_𝐒<R.

In order to write certain equations, we will write, forx∈ 𝐒,x_L^𝐒= 𝐒 ∩x_Landx_R^𝐒≔x_R∩ 𝐒. Notice that we have x= x_L^𝐒|x_R^𝐒 _𝐒by definition. Thus if𝐓is another surreal substructure, the surreal isomorphism Ξ_𝐓^𝐒sendsxonto

Ξ_𝐓^𝐒x= Ξ_𝐓^𝐒x_L^𝐒|Ξ_𝐓^𝐒x_R^𝐒 _𝐓. (1)

Surreal exponentiationGonshor uses an inductive equation to define the exponential func- tion. Givenn∈ ℕandx∈No, set

[x]n=

k⩽n

x^k k!.

Letx∈Nobe such thatexp is defined onx⊏. Notice that for instance, forx′ ∈xLandn∈ ℕ, we should have

exp(x) =exp(x′)exp(x−x′) >exp(x′) [x−x′]n, whereas similar inequalities hold forx′′ ∈x_R. This led Gonshor to define

expx= 0, [x−xL]_ℕexpxL, [x−xR]_2ℕ+1expxR expx_R

[x−xR]_2ℕ+1, expx_L [xL−x]_ℕ .

Gonshor proved that this equation is warranted and that it does define a strictly increasing bijective morphism(No, +) ⟶ (No^>, ×). We writelog for the reciprocal of exp.

The functionshandg The functionexp interacts with theω-map in the following way:

exp(Mo^≻) =Mo^Mo by [9, Chapter 10].

More precisely, for every strictly positive numberxand every numbery, we have exp(ω˙^x) = ω˙^ω˙g(x),

log ω˙^ω˙y = ω˙^h(y),

where the strictly increasing and bijective functiong:No^>⟶Noand its reciprocalhhave the following equations inNo^>andNo[9, Theorems 10.11 and 10.12]:

∀z= {L|R} ∈No^>,g(z) = Ξ_Mo⁻¹ 𝔡z,g(L∩No^>)|g(R) .

∀z= {L|R} ∈No,h(z) = {h(L)|h(R), ℝ^>ω˙^z}_No^>.

The functiongwas entirely studied by Gonshor who gave formal results such as the character- ization of its fixed points and as well as a somewhat informal description of the sign sequence of g(z)for any strictly positive numberzgiven that ofz. We will recover part of his results in a different approach in Section4.

3.2 Log-atomic numbers

The classLaof log-atomic numbers is defined as the class of numbers𝔪 ∈Mo^≻withlogⁿ𝔪 ∈ Mo^≻for alln∈ ℕ. In other words, we have the equality

La≔

n∈ℕ

expⁿMo^≻.

The classLawas first described by Berarducci and Mantova [5] in order to define a derivation onNo. We next describe the parametrization ofLa.

Consider, forr∈ ℝ^>andn∈ ℕthe function

f_n,r≔x⟼expⁿ(rlogⁿz):No^>,≻⟶No^>,≻.

Recall that the class of monomials is that of numbers𝔪which are simplest in each class ℋ[𝔪] = {x∈No^>: ∃r∈ ℝ^>,f_0,r−1(𝔪) <x<f_0,r(𝔪)}.

Similarly, Berarducci and Mantova proved that log-atomic numbersλare exactly the simplest numbers of each class

ℰ[λ] ≔ {x∈No^>: ∃n∈ ℕ, ∃r∈ ℝ^>,f_n,r−1(𝔪) <x<fn,r(𝔪)}.

Moreover, they derived [5, Definition 5.12 and Corollary 5.17] the following equation for the parametrizationλ_⋅≔ΞLaofLa:

∀z∈No,λz = {ℝ,f_ℕ,ℝ^>(λz_L)|f_ℕ,ℝ^>(λz_R)} (2)

= {ℝ,expⁿ(rlogⁿλ_z′)|expⁿ(rlogⁿλ_z′′)}, wherer,n,z′,z′′respectively range inℝ^>,ℕ,zLandzR.

Moreover, the following formula [2, Proposition 2.5] is know forλ_⋅:

∀z∈No,λ_z+1=expλ_z. (3)

3.3 Surreal hyperexponentiation

In [4], we defined a strictly increasing bijectionE_ω:No^>,≻⟶No^>,≻which satisfies the equation

∀x∈No^>,≻,Eω(x+1) =exp(E_ω(x)). (4) Morever, this function is surreal-analytic at every point in the sense of [6, Definition 7.8] and satisfiesEω(x) >E_ℕ(x)for allx∈No^>,≻. This function can be seen as a surreal counterpart to Kneser's transexponential function [12].

In order to defineEω, we relied on a surreal substructureTrof so-called truncated numbers.

They can be characterized as numbers φ∈No^>,≻ withsuppφ≻_L ¹

ℕE_ω(φ). The functionEωis defined on the class of truncated series by the equation:

∀φ∈Tr,Eω(φ) = E_ℕ(φ),f_ℕ,ℝ^> Eω φ_L^Tr |f_ℕ,ℝ^> Eω φ_R^Tr ∈La This equation realizes a strictly increasing bijectionTr⟶La.

Now consider the setNo(ε₀)of surreal numbersxwith lengthℓ(x) <ε₀. By [8, Corollary 5.5], the structure(No(ε0), +, ×,exp)is an elementary extension of the real exponential field. More- over, forx∈No(ε₀), there isn∈ ℕwithx<E_n(ω). It follows that each element ofNo(ε₀)^>,≻≔ No(ε₀) ∩No^>,≻ is truncated, and that for φ∈No(ε₀) ∩No^>,≻, the setsE_ℕ(φ)and λ_ℕ= {ω, exp(ω),exp(exp(ω)), …}are mutually cofinal with respect to one another. Thus onNo(ε₀)^>,≻, the equation forEωbecomes

∀φ∈No(ε₀)^>,≻,Eω(φ) = λ_ℕ,f_ℕ,ℝ^> Eω φ_L^Tr |f_ℕ,ℝ^> Eω φ_R^Tr .

We claim thatEω(φ) =λφfor allφ∈No(ε0). We prove this by induction on(No(ε0)^>,≻, ⊑). Let φ∈No(ε₀)^>,≻such that the result holds onφ_⊏. Noticing thatNo(ε₀)is⊑-initial inNo, we get

E_ω(φ) = λ_ℕ,f_ℕ,ℝ^> E_ω φ_L^Tr |f_ℕ,ℝ^> E_ω φ_R^Tr

= λ_ℕ,f_ℕ,ℝ^> λ_φ

LTr |f_ℕ,ℝ^> λ_φ

RTr by the inductive hypothesis.

= λ_ℕ,f_ℕ,ℝ^> λ_φ

LNo^>,≻ |f_ℕ,ℝ^> λ_φ

RNo^>,≻ becauseNo(ε₀)^>,≻⊆Tr

= {ℝ,f_ℕ,ℝ^>(λφ_L)|f_ℕ,ℝ^>(λφ_R)} becauseφ_L=φ_L^>,≻∪ ℕandφ_R=φ_R^No>,≻

= λφ by (2).

SoE_ωandλ_⋅coincide onNo(ε₀)^>,≻. 3.4 Kappa numbers

The structure𝐊of κ-numbers was introduced first and studied in detail by S. Kuhlmann and M. Matusinski in [13]. It was designed as an intermediate subclass between fundamental mono- mials and log-atomic numbers. The relation between 𝐊 and La is given by the imbrication 𝐊 =La^No≻of [14, p 21].

Similarly to monomials and log-atomic numbers,𝐊numbers can be characterized as simplest elements in each convex class

ℰ^∗[x] ≔ {y∈No^>,≻: ∃n∈ ℕ,logⁿx⩽y⩽expⁿx}, forx∈No^>,≻. The parametrizationκ_⋅≔Ξ_𝐊:No⟶ 𝐊of𝐊is given by the equation

∀z∈No,κz= ℝ,exp^ℕκzL|log^ℕκzR .

Moreover, by [3, Corollary 13 and Theorem 6.16], the structure𝐊is closed.

In order to compute the sign sequence of κ-numbers, Kuhlmann and Matusinski rely on an intermediate surreal substructure denoted𝐈. This surreal substructure is defined by the imbri- cation relation: 𝐊 ≔ Mo²𝐈. Indeed, since𝐊 ⊆La⊆exp(Mo^≻) =Mo², the structure𝐈exists and is unique.

Forz∈Nowe let♭z denote the number withz=1∔ ♭zifz>0 and♭z= −∞ifz⩽0. We also extend the functionsε.andτ to−∞withτ−∞≔ −∞andε−∞≔0. Forz∈No, we defineϕ_z∔1^𝐈 ≔ ετ♭zandϕ_z∔(−1)^𝐈 ≔ −ω. The parametrizationι⋅≔Ξ_𝐈of𝐈admits the following sign sequence for- mula

∀z∈No,ιz= ˙

β<ℓ(z)ϕ_z↿(β+1)^𝐈 [13, Lemma 4.2].

4 Log-atomic numbers and fixed points

Our goal is to compute the sign sequence of ΞLaz=λzin terms of that of z, for all numbersz. Recall thatLa⊆Mo², so there is a surreal substructure𝐑withLa=Mo²𝐑. We writeρ_z≔ Ξ_𝐑z for allz∈No. We have𝐊 =Mo^2I=LaNo≻, whence𝐈 = 𝐑^No≻. The computation in [13] of sign sequences of numbers inLa^(No≻+ ℤ) =exp^ℤ(La^No≻) =log^ℤ𝐊can thus be used to derive part of the result.

Forz=ω×. a+n∈No_≻+ ℤwherea∈Noandn∈ ℤ, we haveλz=expⁿλ_ω×^._a=expⁿκa. By [13, Theorem 4.3], we have

λz = ω˙^ω˙ιa∔n ifn⩽0. (5)

λz = ω˙^{ω˙ιa∔ΞMon (ετ♭a+1)} ifn>0. (6) Our goal is to extend this description to the sign sequences of numbersρzfor allz∈No, relying on the known valuesρω×.a+nfor alla∈Noandn∈ ℤ. More precisely, we will computeρ⋅on all intervals

𝐈a,n: =(ω×.a− (n+1),ω×.a−n) wherea∈Noandn∈ ℤ.

SinceNocan be realized as the reunion No=

(((((((((((((

(( (

(

n∈ℤ

𝐈_a,n

))))))))))))) )) )

)

this will cover all cases. The sign sequence of λz can then be computed using Proposition4 twice. In order to compute the sign sequence for𝐑, we first describe the action of handg on sign sequences in certain cases.

4.1 Computing^h

Leta∈Nobe fixed and writeθa≔ω×.a.

L^EMMA20. We have𝐈 ⊆No≻, whence𝐑^{No≻⊆No≻.}

P^ROOF. The first result is a consequence of the identityNo≻=ω×. Noand [13, Lemma 4.2].

The second immediately follows. □

Recall that we haveκa=ω˙^ω˙ιasoρ_θa=ιa. By [13, Lemma 4.2], we haveτι_a=ε♭τ_a. We set ς_a ≔ τ_ιa∈On and

δa ≔ ω˙^ςa∔1 =ω˙^ςa×.ω∈On,

soδa=ωifτa=0 andδa=ςa×.ωif τa>0. We will treat the casesτa=0 andτa>0 in a uniform way.

L^EMMA21. For n∈ ℕ, we have ρθ_a−n=ρθ_a−n=ρθ_a∔ (−n)

PROOF. We haveρ_θa−n=logⁿρ_θa=logⁿκι_a=ω˙^{ω˙ιa∔(−n)}by [13, Theorem 4.3(1)]. By Lemma20, we haveιa∈No≻soιa∔ (−n) =ιa−n, soω˙^{ω˙ρθa−n}=ρ_θa−n=ω˙^ω˙ιa−n. Sinceρ_θa=ω˙^ω˙ιa, we haveρ_θa−n=

ρ_θa−n. It follows thatρ_θa−n=ρ_θa∔ (−n). □

We now fixn∈ ℕand we set𝐉_a,n≔h(𝐈_a,n). By the previous lemma, we can write the interval 𝐈a,nas the surreal substructure

𝐈a,n = (ρ_θa−n−1,ρ_θa−n)

= (ρ_θa−n−1,ρ_θa−n)

= No^{⊒ρθa−n− /12}.

Thus Ξ_𝐈a,nis straightforward to compute in terms of sign sequences.

We haveh(ρ_θa−n) =ω˙^{ρθa−n−1}andh(ρ_θa−n−1)=ω˙^{ρθa−n−2}=ω˙^{ρθa−n−1}∔ (−δa). Thus the interval 𝐉a,nis subject to the computation given in Lemma9(b). That is, we have:

LEMMA22. For z∈No, we have

a) Ξ_𝐉a,nz=ω˙^{ρθa−n−1}∔ (−1) ∔z if z≮ −(δa)L. b) Ξ_𝐉a,n((−δa) ∔z) =ω˙^{ρθa−n−2}∔1∔z.

Moreover, forx∈𝐈_a,n, we haveω˙^{ρθa−n−1}−1∈xLandω˙^{ρθa−n−1}∈xR. This implies thatω˙^{ρθa−n−2}= h(ρ_θa−n−1) ∈h(xL) and that ω˙^{ρθa−n−1}=h(ρ_θa−n) ∈h(xR). Since with ℝ^>ω˙^x>ω˙^{ρθa−n−2}, we deduce the following equation forhon𝐈_a,n:

∀x∈ 𝐈_a,n,h(x) = ω˙^{ρθa−n−2},h x_L^𝐈a,n |h x_R^𝐈a,n ,ω˙^{ρθa−n−1}

= h x_L^𝐈a,n |h x_R^𝐈a,n _𝐉

a,n.

We see thath↿ 𝐈αis the surreal isomorphism𝐈α⟶ 𝐉α, soh↿ 𝐈α=Ξ_𝐉α∘Ξ_𝐈⁻¹_α . PROPOSITION23. For z∈No, we have

a) h(ρ_θa−n−1∔1∔z) =ω˙^{ρθa−n−1}∔ (−1) ∔z if z≮ −(δa)L. b) h(ρ_θa−n−1∔1∔ (−δa) ∔z) =ω˙^{ρθa−n−2}∔1∔z.

PROOF. This is a direct consequence of the previous lemma and the identity h↿𝐈a,n=Ξ_𝐉^𝐈a,n_a,n. □ L^EMMA24. For z∈ 𝐈a,n, if the number h(z)is a monomial, then there is u∈Nowith δa⊑u and z=ρ_θa−n−1∔1∔ (−δa) ∔u.