The hyperserial field of surreal numbers

For any ordinal

, we show how to define a hyperexponential

and a hyperlogarithm

on the class

of positive infinitely large surreal numbers. Such functions are archetypes of extremely fast and slowly growing functions at infinity. We also show that the surreal numbers form a so-called hyperserial field for our definition.

1Introduction

The ordered field

of surreal numbers was introduced by Conway in [11]. Conway originally used transfinite recursion to define both the surreal numbers (henceforth called numbers), the ordering on No, and the ring operations. For any two sets

and

of numbers with

(i.e.

for all

and

), there exists a number

with

and similar recursive formulas exist for

and for deciding whether

, and

. It is truly remarkable that

turns out to be a totally ordered real-closed field for such “simple” definitions [11]. The bracket

is called the Conway bracket. Using this bracket, we obtain a surreal number in any traditional Dedekind cut, which allows us to embed

into

. In addition,

contains all ordinal numbers

An interesting question is which other operations from calculus can be extended to the surreal numbers. Gonshor has shown how to extend the real exponential function to the surreal numbers [19] and the resulting exponential field

turns out to be elementarily equivalent to

[13]. Berarducci and Mantova recently defined a derivation with respect to

on the surreals [9], again with good model-theoretic properties [2]. In collaboration with Mantova, the authors constructed a surreal solution to the functional equation

which is a bijection of

onto itself [6]. We call

a hyperexponential and its functional inverse

a hyperlogarithm.

The first goal of this paper is to extend the results from [6] to the construction of hyperexponentials

of any ordinal force

, together with their functional inverses

. If

is a successor ordinal, then

satisfies the functional equation

Our second goal is to show that these hyperexponentials are “well-behaved” in the sense that they endow

with the structure of a hyperserial field in the sense of [5].

1.1Motivation and background

Whereas it is natural to study surreal exponentiation and differentiation, it may seem more exotic to define and investigate the properties of surreal hyperexponentials and hyperlogarithms. In fact, the main motivation behind our work is a conjecture by the second author [26, p. 16] and a research program that was laid out in [1] for proving this conjecture. The ultimate goal is to expose the deep connections between two types of mathematical infinities: numerical infinities and growth rates at infinity. Let us briefly recall the rationale behind this connection.

Cantor's ordinal numbers provide us with a way to count beyond all natural numbers and to keep counting beyond the size of any set. However, ordinal arithmetic is rather poor in the sense that we have no subtraction or division and that addition and multiplication do not satisfy the usual laws of arithmetic, such as commutativity. We may regard Conway's surreal numbers as providing a calculus with Cantor's ordinal numbers which does extend the usual calculus with real numbers. In this sense, Conway managed to construct the ultimate framework for computations with numerical infinities.

Another source for computations with infinitely large quantities stems from the study of growth rates of real functions at infinity. The first major results towards a systematic asymptotic calculus of this kind are due to Hardy in [21, 22], based on earlier ideas by du Bois-Reymond [15, 16, 17]. Hardy defined an -function to be a function constructed from

and the real numbers

using the field operations, exponentiation, and logarithms. He proved that the germs of

-functions at infinity form a totally ordered field. The framework of

-functions is suitable for asymptotic analysis since we have an ordering for comparing the growth at infinity of any two such functions. This is often rephrased by saying that

-functions have a regular 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 which may be constructed by means of the logarithmic and exponential functions.” In other words, Hardy suggested that the framework of

-functions not only allows for the development of a systematic asymptotic calculus, but that this framework is also sufficient for all “practical” purposes. Alas, there are several “holes”. First of all, the framework is not closed under various useful operations such as functional inversion and integration. Secondly, the framework does not contain any functions of extremely fast or slow growth at infinity, like

and

, although such functions naturally appear in the analysis of certain algorithms. For instance, the best known algorithm for multiplying two polynomials of degree

runs in time

; see [23].

This raises the question how to construct a truly universal framework for computations with regular functions at infinity. Our next candidate is the class of transseries. A transseries is a formal object that is constructed from

(with

) and the real numbers, using exponentiation, logarithms, and infinite sums. One example of a transseries is

Depending on conditions satisfied by their supports, there are different types of transseries. The first constructions of fields of transseries are due to Dahn and Göring [12] and Écalle [18]. More general constructions were proposed subsequently by the second author and his former student Schmeling [24, 25, 29]. Clearly, any

-function is a transseries, but the class of transseries is also closed under integration and functional inversion, contrary to the class of

-functions.

However, the class of transseries still does not contain any hyperexponential or hyperlogarithmic elements like

. In our quest for a truly universal framework for asymptotic analysis, we are thus lead to look beyond: a hyperseries is a formal object that is constructed from

and the real numbers using exponentiation, logarithms, infinite sums, as well as hyperexponentials

and hyperlogarithms

of any force

. The hyperexponentials

and the hyperlogarithms

are required to satisfy functional equations

It is non-trivial to construct fields of hyperseries in which these and several other technical properties (see section 4 below) are satisfied. This was first accomplished by Schmeling for hyperexponentials

and hyperlogarithms

of finite force

. The general case was tackled in [14, 5].

The construction of general hyperseries relies on the definition of an abstract notion of hyperserial fields. Whereas the hyperseries that we are really after should actually be hyperseries in an infinitely large variable

, abstract hyperserial fields potentially contain hyperseries that can not be written as infinite expressions in

. In the present paper, we define hyperexponentials

and hyperlogarithms

for all ordinals

and show that this provides

with the structure of an abstract hyperserial field. Moreover, any hyperseries

can naturally be evaluated at

to produce a surreal number

. The conjecture from [26, p. 16] states that, for a sufficiently general notion of “hyperseries in

”, all surreal numbers can actually be obtained in this way. We plan to prove this and the conjecture in a follow-up paper.

1.2General overview and summary of our new contributions

Our main goal is to define hyperexponentials

for any ordinal

and to show that

is a hyperserial field for these hyperexponentials. Since our construction builds on quite some previous work, the paper starts with three sections of reminders.

In section 2, we recall basic facts about well-based series and surreal numbers. In particular, we recall that any surreal number

can be regarded as a well-based series

with real coefficients

. The corresponding group of monomials

consists of those positive numbers

that are of the form

for certain subsets

and

with

Section 3 is devoted to the theory of surreal substructures from [4]. One distinctive feature of the class of surreal numbers is that it comes with a partial, well-founded order

, which is called the simplicity relation. The Conway bracket can then be characterized by the fact that, for any sets

and

of surreal numbers with

, there exists a unique

-minimal number

with

. For many interesting subclasses

, it turns out that the restrictions of

and

give rise to a structure

that is isomorphic to

. Such classes

are called surreal substructures of

and they come with their own Conway bracket

In section 4, we recall the definition of hyperserial fields from [5] and the main results on how to construct such fields. One major fact from [5] on which we heavily rely is that the construction of hyperserial fields can be reduced to the construction of hyperserial skeletons. In the context of the present paper, this means that it suffices to define the hyperlogarithms

only for very special, so called -atomic elements.

In the case when

, the

-atomic elements are simply the monomials in

and the definition of the general logarithm on

indeed reduces to the definition of the logarithm on

: given

, we write

, where

and

is infinitesimal, and we take

. This very special case will be considered in more detail in section 5.

In the case when

, the

-atomic elements of

are those elements

such that

is a monomial for every

. The construction of

then reduces to the construction of

on the class

-atomic numbers. This particular case was first dealt with in [6] and this paper can be used as an introduction to the more general results in the present paper.

For general ordinals

, we say that

-atomic if

is a monomial for every

. The advantage of restricting ourselves to such numbers

when defining hyperlogarithms is that

only needs to verify few requirements with respect to the ordering. This makes it possible to define

using the fairly simple recursive formula

In section 6, we prove that this definition is warranted and that the resulting functions

satisfy the axioms of hyperserial skeletons from [5, Section 3]. Our proof proceeds by induction on

and also relies on the fact that the class

-atomic numbers actually forms a surreal substructure of

. Our main result is the following theorem:

Theorem 1.1. The definition (1.5) gives the structure of a confluent hyperserial skeleton in the sense of [5]. Consequently, we may uniquely extend to in a way that gives the structure of a confluent hyperserial field. Moreover, for each ordinal , the extended function is bijective.

Our final section 7 is devoted to further identities that illustrate the interplay between the hyperexponential and hyperlogarithmic functions and the simplicity relation

. We also prove the following more symmetric variant of (1.5):

where

again range over the

-atomic numbers with

and

. 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

2Basic notions

2.1Ordered fields of well-based series

2.1.1Well-based series

Let

be a (possibly class-sized) linearly ordered abelian group. We write

for the class of functions

whose support

is a well-based set, i.e. a set which is well-ordered with respect to the reverse order

We see elements

as formal well-based series

, where

denotes the coefficient

, for each

. If

, then we define

to be the dominant monomial of

. For

, we let

and we write

. We say that a series

is a truncation of

and we write

. The relation

is a well-founded partial order on

with minimum

The identification of

with the formal series

induces an ordered group embedding

The relation

extends the ordering on

. For non-zero

we actually have

(resp.

, resp.

) if and only if

(resp.

, resp.

). We finally define

Series in

and

are respectively called purely large, infinitesimal, and positive infinite.

2.1.2Well-based families

is another field of well-based series and

-linear, then we say that

is strongly linear if for every well-based family

, the family

is well-based, with

2.2Surreal numbers

2.2.1Surreal numbers and simplicity

We denote by

the class of ordinal numbers. Following [19], we define

to be the class of sign sequences

of ordinal length

. The terms

are called the signs of

and we write

for the length of

. Given two numbers

, we define

We call

the simplicity relation on

and note that

is well-founded. See [4, Section 2] for more details about the interaction between

and the ordered field structure of

Recall that the Conway bracket is characterized by the fact that, for any sets

and

of surreal numbers with

, there exists a unique

-minimal number

with

. Conversely, given a number

, we define

2.2.2Ordinals as surreal numbers

The structure

contains an isomorphic copy of

by identifying each ordinal

with the constant sequence

of length

. We will write

to state that

is either an ordinal or the class of ordinals.

is a successor ordinal, then we define

to be the unique ordinal with

. We also define

is a limit ordinal. Similarly, if

, then we set

. Recall that every ordinal

has a unique Cantor normal form

2.2.3Surreal numbers as well-based series

We define

to be the class of positive numbers

of the form

for certain subsets

and

with

. Numbers in

are called monomials. It turns out [11, Theorem 21] that the monomials form a subgroup of

and that there is a natural isomorphism between

and the ordered field

. We will identify those two fields and thus see

as a field of well-based series. The ordinal

, seen as a surreal number, is the simplest element, or

-minimum, of the class

3Surreal substructures

3.1Surreal substructures

In [4], we introduced the notion of surreal substructures. A surreal substructure is a subclass

such that

and

are isomorphic. The isomorphism

is unique and denoted by

. Many important subclasses of

that are relevant to the study of hyperserial properties of

are surreal substructures. In particular, it is known that the following classes are surreal substructures:

3.2Cuts

is called a cut in

. If

contains a unique simplest element, then we denote this element by

and say that

is a cut representation (of

) in

. These notations naturally extend to the case when

and

are subclasses of

with

A surreal substructure

may be characterized as a subclass of

such that for all cut representations

, the cut

has a unique simplest element [4, Proposition 4.7].

Let

be a surreal substructure. Note that we have

for all

. Let

and let

be a cut representation of

. Then

is cofinal with respect to

in the sense that

has no strict upper bound in

and

has no strict lower bound in

[4, Proposition 4.11(b)].

Given numbers

with

, the number

is the unique

-maximal number with

. We have

. Let

be a surreal substructure. Considering the isomorphism

, we see that for all

with

, there is a unique

-maximal element

with

, and we have

. In what follows, we will use this basic fact several times without further mention.

3.3Cut equations

Let

be a subclass, let

be a surreal substructure and

be a function. Let

be functions defined for cut representations in

and such that

are subsets of

whenever

is a cut representation in

. We say that

is a cut equation for

if for all

, we have

Elements in

(resp.

) are called left (resp. right) options of this cut equation at

whenever

is a cut representation in

. For instance, given

, consider the translation

. By [19, Theorem 3.2], we have the following uniform cut equation for

Proposition 3.1. [4, Proposition 4.36] Let be surreal substructures. Let be a function from to the class of subsets of such that for with , the set is cofinal with respect to . For , let denote the class of elements of such that and are mutually cofinal. Let be a cut equation on that is extensive in the sense that

3.4Convex partitions

One natural way to obtain surreal substructures is via convex partitions. If

is a surreal substructure, then a convex partition of

is a partition

whose members are convex subclasses of

for the order

. We may then consider the class

of simplest elements (i.e.

-minima) in each member of

. Those elements are said

-simple. For

, we let

denote the unique member of

containing

. By [4, Proposition 4.16], the class

contains a unique

-simple element, which we denote by

. The function

is a surjective non-decreasing function

with

Given

, note that we have

if and only if

. For

, we write

. We have the following criterion to characterize elements of

Proposition 3.2. [4, Lemma 6.5] An element of is -simple if and only if there is a cut representation of in with . Equivalently is -simple if and only if .

We say that

is thin if each member of

has a cofinal and coinitial subset. We then have:

Proposition 3.3. [4, Theorem 6.7 and Proposition 6.8] If is thin, then the class is a surreal substructure and has the following uniform cut equation:

3.5Function groups

A special type of thin convex partitions is that of partitions induced by function groups acting on surreal substructures. A function group

on a surreal substructure

is a set-sized group of strictly increasing bijections

under functional composition. We see elements

as actions on

and we sometimes write

and

instead of

and

, where

with

is a thin convex partition of

. We write

. We have the uniform cut equation

Consider sets

of strictly increasing bijections

, then we say that

is pointwise cofinal with respect to

, and we write

, if we have

. We also define

It is easy to see that

is a function group on

and that we have

. The relation

trivially implies

. If

and

, then we say that

and

are mutually pointwise cofinal and we write

. We then have

Given a function group

, the relation defined by

is a partial order on

. We will frequently rely on the basic fact that

is partially bi-ordered in the sense that

3.6Remarkable function groups

Each of the examples of surreal substructures from Subsection 3.1 can be regarded as the classes

for actions of the following function groups

acting on

. For

and

, we define

Generalizations of those function groups will allow us to define certain surreal substructures related to the hyperlogarithms and hyperexponentials on

4Hyperserial fields

In this section, we briefly recall the definition of hyperserial fields from [5] and how to construct such fields from their hyperserial skeletons.

4.1Logarithmic hyperseries

Let

be a formal, infinitely large indeterminate. The field

of logarithmic hyperseries of [14] is the smallest field of well-based series that contains all ordinal real power products of the hyperlogarithms

with

. It is naturally equipped with a derivation

and composition law

Definition

We define

to be the ordered field of well-based series

. If

are ordinals with

, then we define

to be the subgroup of

of monomials

with

whenever

. As in [14], we write

Derivation on

Composition on

The same properties hold for the composition

is replaced by

. For

, the map

is injective, with image

[14, Lemma 5.11]. For

, we define

to be the unique series in

with

4.2Hyperserial fields

of ordered

-vector space on

. Let

be a field of well-based series with

, let

, and let

be a function. For

, we define

to be the class of series

with

. We say that

is a hyperserial field if

For each

, we define the function

. The skeleton of

is defined to be the structure

equipped with the real power operation from HF5.

4.3Hyperserial skeletons

It turns out that each hyperlogarithm

on a hyperserial field

can uniquely be reconstructed from its restriction to the subset of

-atomic hyperseries (here we say that

-atomic if

for all

). One of the main ideas behind [14] is to turn this fact into a way to construct hyperserial fields. This leads to the definition of a hyperserial skeleton as a field

with partially defined hyperlogarithms

, which satisfy suitable counterparts of the above axioms HF1 until HF7.

More precisely, let

be a field of well-based series and fix

. A hyperserial skeleton on

of force

consists of a family of partial functions

for

, called (hyper)logarithms, which satisfy a list of axioms that we will describe now.

First of all, the domains

on which the partial functions

are defined should satisfy the following axioms:

Consider an ordinal

written in Cantor normal form

where

and

. We denote by

the partial function

It follows from the definition that for all

, the class

consists of those series

for which

and

for all

. We call such series

-atomic.

Note that

and

together imply

, whence

automatically holds. This will in particular be the case for

(see Section 5).

Definition 4.1. [5, Definition 3.3] Given , we say that is a hyperserial skeleton of force if it satisfies , , , , and for all , as well as for all ordinals .

Assume that

is a hyperserial skeleton of force

. The partial logarithm

extends naturally into a strictly increasing morphism

, which we call the logarithm and denote by

[5, Section 4.1]. If

satisfies

, then this extended logarithm is actually an isomorphism [29, Proposition 2.3.8]. In that case, for any

and

, we define

4.4Confluence

Definition 4.2. [5, Definition 3.5] Given a hyperserial skeleton of force and , we inductively define the notion of -confluence in conjunction with the definition of functions , as follows.

We say that is -confluent if each class contains a -atomic element; we then define to be this element.

This inductive definition is sound. Indeed, if

and

-confluent for all

, then the functions

with

are well-defined and non-decreasing. Thus, for

, the collection of

with

forms a partition of

into convex subclasses.

We say that

is confluent if it is

-confluent. If

has force

, then we say that

-confluent, or confluent, if

-confluent for all

4.5Correspondence between fields and skeletons

Proposition 4.3. [5, Theorem 1.1] If is a confluent hyperserial skeleton, then there is a unique function with

Assume now that

is only a hyperserial skeleton of force

and that

is an ordinal with

such that

-confluent. Let

. By [5, Definition 4.11 and Lemma 4.12], the partial function

naturally extends into a function

that we still denote by

. This extended function is strictly increasing, by‘ [5, Corollary 4.17]. If

is a successor ordinal, then it satisfies the functional equation

by [5, Proposition 4.13]. For

, we have a strictly increasing function

obtained as a composition of functions

with

, as in (4.1). By [5, Proposition 4.7], we have

4.6Hyperexponentiation

In a traditional transseries field

, the transmonomials are characterized by the fact that, for any

, we have

In particular, the logarithm

is surjective as soon as

is defined for all

with

. In hyperserial fields, similar properties hold for

-atomic elements with respect to the hyperexponential

, as we will recall now.

Given

, let

be a confluent hyperserial skeleton

of force

. By [5, Theorem 4.1], we have a composition

. Given

, the extended function

is strictly increasing and hence injective. Consequently,

has a partially defined functional inverse that we denote by

Let

be a series such that

is defined. By [5, Lemma 7.14], the series

-truncated if and only if

For

, the axiom

is therefore equivalent to the inclusion

. For

, there is a unique

-maximal truncation

which is

-truncated. By [5, Propositions 6.16 and 6.17], the classes

with

form a partition of

into convex subclasses. Moreover, the series

is both the unique

-truncated element and the

-minimum of

. If

is defined, then we have the following simplified definition [5, Proposition 7.19] of the class

The following shows that the existence of

is essentially equivalent to its existence on

Proposition 4.6. [5, Corollary 7.24] Let and assume that for , the function is defined on . Then each hyperlogarithm for is bijective.

If Proposition 4.6 holds, then we say that

is a (confluent) hyperserial field of force

. Since every function

is then a strictly increasing bijection

, we obtain

5The transseries field

Recall that

is identified with the ordered field of well-based series

. In this section, we describe, in the first level

of our hierarchy, the properties of

equipped with the Kruskal-Gonshor logarithm.

5.1Surreal exponentiation

In [19, Chapter 10], Gonshor defines the exponential function

, relying on partial Taylor sums of the real exponential function. For

and

, write

We will sometimes write

instead of

. The function

is a bijective morphism [19, Corollary 10.1, Corollary 10.3], which satisfies:

We define

to be the functional inverse of

, and we set

. Given an ordinal

, we understand that

still stands for the

-th ordinal power of

from section 2.2.2 and warn the reader that

does not necessarily coincide with

Together, the above facts imply that

satisfies the axioms

and

. Therefore,

is a hyperserial skeleton of force

. The extension of

from section 4.5 coincides with

. It was shown in [13] that

is an elementary extension of

. See [28, 7, 8] for more details on

and

5.2

as a transseries field

Berarducci and Mantova identified the class

-atomic numbers as

[9, Corollary 5.17] and showed that

-confluent [9, Corollary 5.11]. Thus

is a confluent hyperserial skeleton of force

. Thanks to [5, Theorem 1.1], it is therefore equipped with a composition law

. See [29, 10] for further details on extensions of this composition law to exponential extensions of

Berarducci and Mantova also proved [9, Theorem 8.10] that

is a field of transseries in the sense of [24, 29], i.e. that

satisfies the axiom

of [29, Definition 2.2.1]. We plan to prove in subsequent work that

satisfies a generalized version of

6Hyperserial structure on

We have seen in section 5 that

is a confluent hyperserial skeleton of force

for

. The aim of this section is to extend this result to any ordinal

. More precisely, we will define a sequence

of partial functions on

such that for each ordinal

, the structure

is a confluent hyperserial skeleton of force

, and

coincides with Gonshor's logarithm.

6.1Remarkable group actions on

Assume for the moment that we can define

and

as bijective strictly increasing functions on

for all ordinals

. This is the case already for

. Let us introduce several useful groups that act on

, as well as several remarkable subclasses of

By Proposition 3.3 and the fact the set-sized function groups

, and

induce thin partitions of

, we may define the following surreal substructures

Here we note that

corresponds to the class

of infinite monomials in

and

maps positive infinite numbers to their dominant monomial. Similarly,

coincides with

and

maps

. In sections 6 and 7, we will prove the following identities.

The first and third identities imply in particular that the classes

and

from section 4 are in fact surreal substructures, when regarding

as a hyperserial field.

6.2Inductive setting

For the definition of the partial hyperlogarithm

, we will proceed by induction on

. Let

be an ordinal. Until the end of this section we make the following induction hypotheses:

These induction hypotheses require a few additional explanations. Assuming that

holds, the partial functions

with

extend into strictly increasing bijections

, by the results from section 4. Using (1.3), this allows us to define a strictly increasing bijection

for any

and we denote by

its functional inverse. In particular, this ensures that the hypotheses

and

make sense.

Remark 6.1. In addition to the above induction hypotheses, we will implicitly assume that our hyperlogarithms

for

are always defined by (6.1) below. In particular, our construction of

is not relative to any potential construction of the preceding hyperlogarithms

with

that would satify the induction hypotheses

, and

. Instead, we define one specific family of functions

that satisfy our requirements, as well as the additional identities listed in Section 6.1.

Proposition 6.3. Let be a limit ordinal and assume that , , and hold for all . Then , , and hold.

Proof. The statement

follows immediately by induction. Towards

, note that we have

(and thus

) and

for all

. By [4, Proposition 6.28], we have

. So

holds.

The remainder of the section is dedicated to the definition of

and the proof of the inductive hypotheses I, I, and I for

. In combination with Propositions 6.2 and 6.3, this will complete our induction and the proof of Theorem 1.1.

6.3Defining the hyperlogarithm

Recall that we have

. In particular

is a surreal substructure. Consider

. The skeleton

is a confluent hyperserial skeleton of force

. So for

, (4.7) and

yield

Note the asymmetry between left and right options

and

(instead of

) for generic

and

. In Corollary 7.4 below, we will derive a more symmetric but equivalent cut equation for

, as promised in the introduction. For now, we prove that (6.1) is warranted and that

, and

hold.

Proof. We prove this by induction on

. Let

such that

holds for all

. Let

and

. We have

, so

yields

Proof. Let

and write

. Let

with

. We have

and

. Moreover

is positive infinite. The number

is strictly simpler than

, so

does not lie in the cut which defines

in (6.1). Therefore, there is an

or an

and an ordinal

with

. Consider the first case. We have

for a certain

. So

and

Proposition 6.8. If is a successor ordinal, then the cut equation (6.1) is uniform.

Proof. Let

be a cut representation in

and write

. For

, we have

. For

, we have

. Since

, it follows that

. We may thus define the number

In order to show that (6.1) is uniform, we need to prove that

, for any choice of the cut representation

. We will do so by proving that

and

Recall that

is cofinal with respect to

and that

is strictly increasing. Consequently, we have

Given

, there is an

with

. By

, we have

for all

, so

. This proves that

lies in the cut defining

as per (6.1), whence

Let

and let

-maximal with

. We have

, whence

, by

. If

, then

, so

yields

and

. Otherwise

, so

yields

. This proves that

6.4Functional equation

In this subsection we derive

, under the assumption that

is a successor ordinal. We start with the following inequality.

. Symmetric arguments yield

. Lemma 6.9 implies that

, whence

. We get

, whence

. Thus

lies in the cut defining

in (6.2), so

. This proves our claim that

Proof. We prove this by induction on

. Let

be such that the result holds on

. By (6.3), we have

Let

and

range in

and

respectively. Proposition 6.8 and our induction hypothesis yield:

Combining our results so far, we have proved that

is a hyperserial skeleton of force

6.5Confluence

Lemma 6.11. If is a non-zero limit ordinal, then the function groups and are mutually pointwise cofinal. In particular, we have and .

Proof. Let

denote the class of numbers

that are

-minimal in

. Any such

-minimal number

is also

-minimal in

, hence

-atomic. Thus

is defined on

. It is enough to prove that

is closed under

in order to obtain that

Assume for contradiction that

is not

-minimal in

. So there is a

with

. This implies that

lies outside the cut defining

, so

is larger than a right option of (6.4) or smaller than a left option of (6.4).

Now consider the other case when

. In particular,

must be larger than a right option of (6.4). Symmetric arguments imply that we cannot have

for some

. So there must exist a

with

. If

is a limit ordinal, then

so Lemma 6.11 yields

, whence

. If

is a successor ordinal, then there is a

with

, so

and Proposition 6.10 yields

. In both cases, we thus have

. For any integer

, we deduce that

The corollary implies that

is a confluent hyperserial skeleton of force

. Moreover, the class

is that of

-minima and thus

-minima in the convex classes

for

. In other words, we have

. In order to conclude that

is a surreal substructure, we still need to prove that the convex partition

is thin. This will be done at the end of section 6.6 below.

6.6Hyperexponentials

We have shown that

is a hyperserial skeleton of force

. In order to prove that

has force

, it remains to prove that every

-truncated number

has a hyperexponential

. This is the purpose of this subsection.

Proof. We prove the result by induction on

. Let

such that

is defined on

with the given equation. We will first show that the number

Let

and

. If

, then

by the definition of

. So

. Otherwise, we have

, whence

by definition of

, so

. So we always have

Let us now prove that

. Note that

by Proposition 3.2. First assume that

is a limit ordinal. Lemma 6.11 yields

, so we may write

By (4.6), for

the classes that

and

are mutually cofinal and coinitial. Moreover, we have

for all

, by our hypothesis on

. Hence, Proposition 6.14 and (4.6) imply

. Since

, the inequality

follows from Proposition 3.2. Finally, we have by definition that

, so

. This proves that

, so

Assume now that

is a successor ordinal. For all

, the sets

, and

are mutually cofinal. So we can rewrite (6.6) as

With Proposition 6.15, we have completed the proof of

. By (4.7), we have

for all

. Given

, we also deduce from (4.6) that the set

is cofinal and coinitial in

. The convex partition defined by

is thus thin. By Proposition 3.3, the class

is a surreal substructure with uniform cut equation

For

, we have

, so

. We deduce that the following equivalent is equivalent to (6.8):

6.7End of the inductive proof

Proof. Throughout this proof, we consider inequalities and equalities of functions on

. Write

and

where

and

. We have

Proof. Let

. By Corollary 6.13, the simplest element of

-atomic. Since

, we deduce that

In particular, the class

is a surreal substructure. We have proved

, and

, so we obtain the following by induction:

Combining this with Propositions 4.3 and 4.6, we obtain Theorem 1.1. Let us finally show that

contains only one

-atomic element.

Proposition 6.20. The number is the only -atomic element in . For all , there is with .

7Remarkable identities

In this section, we give various identities regarding the function groups introduced in Section 6.1. In what follows,

is a non-zero ordinal and

7.1Simplified cut equations for

and

Given

, let

is a successor ordinal and

is a limit ordinal. In this subsection, we will derive the following simplified cut equations for

and

For all

, the set

contains only

-atomic numbers, so (7.3) is indeed a cut equation of the form

Remark 7.1. The changes with respect to (6.1) and (6.5) lie in the occurrence of

instead of

in (7.2) and the (related) absence of the left option

in (7.4). So (7.2) and (7.4) give lighter sets of conditions than those in (6.1) and (6.5) to define

and

. This seemingly meager simplification will be crucial in further work. Indeed, combined with Proposition 3.1, this allows one to determine large classes of numbers

with

First note that the cut equations (7.1) and (7.3) if they hold are uniform (see [6, Remark 1]). Moreover, we claim that (7.1,7.2) are equivalent and that (7.3,7.4) are equivalent. Indeed, recall that for a thin convex partition

of a surreal substructure

and any cut representation

, one has

For

and

the classes

and

are mutually cofinal by (4.6). Similarly,

and

are mutually coinitial. By Lemma 6.11, the classes

and

are mutually cofinal. So it is enough to prove that (7.1) and (7.3) are valid cut equations for

and

respectively.

Lemma 7.2. If is a successor ordinal, then the identities (7.1) and (7.3) hold.

We have

so in view of (6.1), it is enough to prove that

to conclude that

. Let

. If

, then the inequality

entails

whence

and

. Otherwise, we have

, so

, and

. It is enough to prove that

. Recall that

Proof. We prove the result by induction on

. Let

be such that (7.5), (7.6), (7.7) and (7.8) hold for all

with

For

and

, we have

. We have

by definition of

and by definition of

. This proves that

is defined.

Let

and

. If

, then we have

by definition of

. Since

and

, we have

for all

We deduce that

. If

, then

by definition of

. Since

, we obtain

. This proves that

is defined.

From (7.7), we get

. By Proposition 6.14 and our assumption that (7.8) holds on

, we have

Corollary 7.4. The identities (7.1), (7.2), (7.3), and (7.4) all hold.

Proof. It is enough to prove (7.1) and (7.3). The identity (7.3) follows from (7.7) and (7.8). In order to obtain (7.1), we consider

, set

, and we show that

. Since (7.3) is uniform, we have

Remark 7.5. The simplified cut equations for

can be viewed as alternative definitions for those functions, since they hold inductively on their domain of definition. It is unclear how to develop our theory directly upon these alternative definitions. In particular, does there exists a direct way to see that the cut equation (7.2) is warranted, and that the corresponding function satisfies

and

7.2Identities involving

and

Assume that

is a successor ordinal. Then we have

by (4.4), so the functions

and

are both strictly increasing bijections from

onto

whenever

is strictly simpler than

. We let

denote generic elements of

and we note that

. By (6.8), we have

Recall that

is a successor ordinal. Since (4.2) holds for all

, the sets

and

are mutually cofinal and coinitial. Moreover

for all

and

, so

Proof. The set

is pointwise cofinal in

. So

is pointwise cofinal in

. For

, there is

such that

. We have

7.3Identities involving

and

Proof. This can be seen as a converse to the proof of the identity (6.3). We proceed by induction on

. Let

be such that the relation holds on

. By (6.3), we have

for all strictly simpler

with respect to the product order

. For

, let

be the function

and let

. By (3.1) and (3.2), we have

It is enough to prove that

to conclude that

. Towards this, fix an

with

. Lemma 7.9 yields

Remark 7.11. For

, we have

, and

. Therefore we can see

as a system of fractional and real iterates of the hyperexponential function

. The previous proposition shows that the action of those iterates on

-atomic numbers reduces to translations, modulo the parametrization

. In particular, one can compute the functional square root of

in terms of sign sequences using the material from [3].

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