
A. vincent.bagayoko@umons.ac.be
B. vdhoeven@lix.polytechnique.fr
. This article has been written using GNU TeXmacs [27].
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 socalled hyperserial field for our definition.
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 all numbers can be obtained in this way. Given and , we have
and similar recursive formulas exist for , and for deciding whether , , and . It is truly remarkable that turns out to be a totally ordered realclosed 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
so is actually a proper class.
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 modeltheoretic 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 “wellbehaved” in the sense that they endow with the structure of a hyperserial field in the sense of [5].
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 BoisReymond [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 in 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 or . 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
where . For in Cantor normal form with , we also define
and we require that
It is nontrivial 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 on for all ordinals and show that this provides with the structure of an abstract hyperserial field. Moreover, any hyperseries in 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 followup paper.
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 wellbased series and surreal numbers. In particular, we recall that any surreal number can be regarded as a wellbased series
with real coefficients . The corresponding group of monomials consists of those positive numbers that are of the form for certain subsets and of 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, wellfounded 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 of , it turns out that the restrictions of and to 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 on then reduces to the construction of on the class of 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 is 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
where range over atomic numbers with and ; see also (7.1).
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 of atomic numbers actually forms a surreal substructure of . Our main result is the following theorem:
Our final section 7 is devoted to further identities that illustrate the interplay between the hyperexponential and hyperlogarithmic functions and the simplicity relation on . 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 .
Let be a (possibly classsized) linearly ordered abelian group. We write for the class of functions whose support
is a wellbased set, i.e. a set which is wellordered with respect to the reverse order .
We see elements of as formal wellbased series , where denotes the coefficient of in , 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 if . The relation is a wellfounded partial order on with minimum .
By [20], the class is an ordered field under the pointwise sum
the Cauchy product
(where each sum has finite support), and where the positive cone is given by
The identification of with the formal series induces an ordered group embedding .
We next define the following asymptotic relations on :
The relation extends the ordering on . For nonzero 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.
Let be a family in , We say that is wellbased if
is wellbased, and
is finite for all .
In that case, we may define the sum of by
If is another field of wellbased series and is linear, then we say that is strongly linear if for every wellbased family in , the family is wellbased, with
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 wellfounded. 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
Then can canonically be written as
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.
For , we write for the ordinal exponentiation of to the power and we define
If is a successor ordinal, then we define to be the unique ordinal with . We also define if is a limit ordinal. Similarly, if , then we set . Recall that every ordinal has a unique Cantor normal form
where , and with .
We define to be the class of positive numbers of the form for certain subsets and of 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 wellbased series. The ordinal , seen as a surreal number, is the simplest element, or minimum, of the class .
In [4], we introduced the notion of surreal substructures. A surreal substructure is a subclass of 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:
The classes , and of positive, positive infinite and infinitesimal numbers.
The classes and of monomials and infinite monomials.
The classes and of purely infinite and positive purely infinite numbers.
The class of logatomic numbers.
If are surreal substructures, then the class is a surreal substructure with .
Given a subclass of and , we will write
so that and . We also write and .
If is a subclass of and are subsets of with , then the class
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 in , 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 in . 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 of with , and we have . In what follows, we will use this basic fact several times without further mention.
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 .
We say that the cut equation is uniform if we have
whenever is a cut representation in . For instance, given , consider the translation on . By [19, Theorem 3.2], we have the following uniform cut equation for on :
(3.1) 
We will need the following result from [4]:
Let be strictly increasing with cut equation
Then induces an embedding for each element of .
One natural way to obtain surreal substructures is via convex partitions. If is a surreal substructure, then a convex partition of is a partition of 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 nondecreasing function with .
Given , note that we have if and only if . For , we write . We have the following criterion to characterize elements of .
We say that is thin if each member of has a cofinal and coinitial subset. We then have:
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 setsized group of strictly increasing bijections under functional composition. We see elements of as actions on and we sometimes write and instead of and , where .
For such a function group , the collection of classes
with is a thin convex partition of . We write . We have the uniform cut equation
(3.2) 
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 if or . The relation trivially implies . If and , then we say that and are mutually pointwise cofinal and we write . We then have .
We write (resp. ) if we have (resp. ). We also write and instead of and .
Given a function group on , the relation defined by is a partial order on . We will frequently rely on the basic fact that is partially biordered in the sense that
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 , or . For and , we define
Then we have the following list of correspondences :
The action of on (resp. ) yields (resp. ), e.g. .
The action of on (resp. ) yields (resp. ).
The action of on yields .
The action of on yields .
The action of on yields (which will coincide with ).
Generalizations of those function groups will allow us to define certain surreal substructures related to the hyperlogarithms and hyperexponentials on .
In this section, we briefly recall the definition of hyperserial fields from [5] and how to construct such fields from their hyperserial skeletons.
Let be a formal, infinitely large indeterminate. The field of logarithmic hyperseries of [14] is the smallest field of wellbased series that contains all ordinal real power products of the hyperlogarithms with . It is naturally equipped with a derivation and composition law .
We define to be the ordered field of wellbased series . If are ordinals with , then we define to be the subgroup of of monomials with whenever . As in [14], we write
We have natural inclusions , hence natural inclusions .
So for all . For and , we will sometimes write .
For , the map is a strongly linear embedding [14, Lemma 6.6].
For and , we have and [14, Proposition 7.14].
For and successor ordinals , we have [14, Lemma 5.6].
The same properties hold for the composition if is replaced by . For , the map is injective, with image [14, Lemma 5.11]. For , we define to be the unique series in with .
Let be an ordered group. A real powering operation on is a law
of ordered vector space on . Let be a field of wellbased 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 all , , and .
for all , , and with .
for all ordinals , , and with .
The map extends to a real powering operation on .
for all .
for all , and .
For each , we define the function . The skeleton of is defined to be the structure equipped with the real power operation from HF5.
We say that is confluent if for all with , we have
In particular is a confluent hyperserial field.
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 is 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 wellbased 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:
It will be convenient to also define the class by
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.
Secondly, the hyperlogarithms with should satisfy the following axioms:
Axioms for the logarithm Functional equation: Asymptotics: Monotonicity: Regularity: Surjective logarithm: 
Axioms for the hyperlogarithms (for each with and ) 
Finally, for with , we also need the following axiom
Note that and together imply , whence automatically holds. This will in particular be the case for (see Section 5).
In summary, we have:
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 or [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 .
The field is said confluent if is nontrivial. The function maps every positive infinite series onto its dominant monomial . For each , we write
Let be such that is confluent for all and let .
If is a successor ordinal, then we write for the class of series with
for a certain .
If is a limit ordinal, then we write for the class of series with
for a certain .
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 is confluent for all , then the functions with are welldefined and nondecreasing. 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 is confluent, or confluent, if is confluent for all .
such that is a confluent hyperserial field.
Assume now that is only a hyperserial skeleton of force and that is an ordinal with such that is 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
(4.2) 
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
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 .
The characterization (4.3) generalizes as follows:
Given , we say that a series is truncated if
For any , we write for the class of truncated series in .
In general, we have . Whenever is a successor ordinal, we even have
Let be a series such that is defined. By [5, Lemma 7.14], the series is truncated if and only if
For , the axiom is therefore equivalent to the inclusion . For , there is a unique maximal truncation of 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 on is essentially equivalent to its existence on .
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
for each ordinal with . By [5, Corollary 7.23], for all , we have
Recall that is identified with the ordered field of wellbased series . In this section, we describe, in the first level of our hierarchy, the properties of equipped with the KruskalGonshor logarithm.
In [19, Chapter 10], Gonshor defines the exponential function , relying on partial Taylor sums of the real exponential function. For and , write
We then have the recursive definition
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 to 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 .
Berarducci and Mantova identified the class of atomic numbers as [9, Corollary 5.17] and showed that is 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 .
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.
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 .
Given an ordinal , we write and we consider the function groups
where , and act on . We also define
We write and for each . In the case when , note that
By Proposition 3.3 and the fact the setsized 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 to . 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.
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:
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
From now on, we assume that , , and are satisfied for and we define
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.
Recall that we have by . In particular is a surreal substructure. Consider . The skeleton is a confluent hyperserial skeleton of force by . So for , (4.7) and yield .
In view of and , the simplest way to define is via the cut equation:
(6.1) 
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.
.
For , we have and , whence
for all . Hence,
We clearly have . Finally,
so . This shows that is defined and
Since , it follows that
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 .
Conversely, in order to prove that , it suffices to show that lies in the cut
Let and let be maximal with . We have , whence , by . If , then , so yields and . Otherwise , so yields . This proves that .
In this subsection we derive , under the assumption that is a successor ordinal. We start with the following inequality.
on by (4.2). Note that , so yields
Let . Since is a surreal substructure, we may consider the atomic number
We claim that . Assume that and write . We have
Assume now that . The function is strictly increasing with . Therefore
so . Since , the cut equation (6.1) for yields
Given , we have and . We deduce that
Moreover, by definition, we have
so . 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
(6.3) 
Let and range in and respectively. Proposition 6.8 and our induction hypothesis yield:
On the other hand, we have
Combining our results so far, we have proved that is a hyperserial skeleton of force .
We next prove that is confluent.
Therefore . For , there is with . By (4.2), we have
Consider , and recall that we have
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).
Assume first that . So there is an with . We have so there is an with
The function is nondecreasing, so is nondecreasing as well. So . But
Thus
This contradicts the minimality of .
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
This contradicts the fact that lies in .
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.
By (4.6), this shows that
In particular, the number
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.
(6.5) 
is welldefined. We will then prove that .
Let and . If , then by the definition of . So . Otherwise, we have , whence by definition of , so . So we always have
We also have , so . This proves that . It remains to show that
Note that , so by the definition of , we have
(6.7) 
Hence , which completes the proof that is welldefined.
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
Note that , so . Now . We also have
where
So . 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
As in the limit case, Proposition 6.14 yields
Let . There is an with . Since , we have
In particular . We saw in (6.7) that , whence . We also obtain the inequalities
in a similar way as in the limit case.
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
(6.8) 
For , we have , so . We deduce that the following equivalent is equivalent to (6.8):
We now prove , and Theorem 1.1.
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.
In this section, we give various identities regarding the function groups introduced in Section 6.1. In what follows, is a nonzero ordinal and .
Given , let if is a successor ordinal and if is a limit ordinal. In this subsection, we will derive the following simplified cut equations for on and on :
For all , the set contains only atomic numbers, so (7.3) is indeed a cut equation of the form .
Remark
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 in , 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.
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
by (3.1). We see that for all . We have so . Thus . So (7.1) holds.
Now let and set
By uniformity of (7.1), we have
We now assume that is a limit ordinal. For , define
For and , we have . We have by definition of if and by definition of if . 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.
Since (7.7) and (7.8) hold on , we have
By (6.9), this yields , so (7.7) holds for .
From (7.7), we get . By Proposition 6.14 and our assumption that (7.8) holds on , we have
Remark
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
By (3.1), we have
The numbers and are truncated so lies in the cut
Noting that on , the previous relation further generalizes as follows.
for all strictly simpler with respect to the product order . For , let be the function on and let . By (3.1) and (3.2), we have
By (7.1), Lemma 7.7 and (3.1), we have:
We deduce that
It is enough to prove that to conclude that . Towards this, fix an with . Lemma 7.9 yields
Remark
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simplest number between and 4
field of wellbased series with real coefficients over 5
support of a series 5
truncation of 5
and 5
series with 5
series with 5
series with and 5
class of ordinals 6
simplicity relation 6
ordinal exponentiation with base at 6
if is a successor ordinal and if is a limit ordinal 6
for 6
the surreal substructure 7
class of simple elements 8
projection 8
class of numbers with 8
comparison between sets of strictly increasing bijections 9
function group generated by 9
and are mutually pointwise cofinal 9
translation 9
homothety 9
power function 9
function group 9
function group 9
function group 9
function group 9
function group 9
field of logarithmic hyperseries 10
group of logarithmic hypermonomials of force 10
field of logarithmic hyperseries of force 10
unique series in with 10
hyperlogarithm function 11
class of atomic series 11
functional equation 12
asymptotics axiom 12
monotonicity axiom 12
regularity axiom 12
infinite products axiom 12
class of series with 13
atomic element of 13
hyperexponential function 14
class of truncated series 14
series with 14
maximal truncated truncation of 14
function group 16
function group 16
function group 16
function group 16
structure of simple elements 16
structure of simple elements 16
structure of simple elements 16
structure of simple elements 16