Surreal numbers as hyperseries

Vincent Bagayoko

Département de Mathématique

Université de Mons, Le Pentagone

20, Place du Parc

B-7000 Mons, Belgique

Email: vincent.bagayoko@umons.ac.be

Joris van der Hoeven

CNRS, LIX

Campus de l'École polytechnique

1, rue Honoré d'Estienne d'Orves

Bâtiment Alan Turing, CS35003

91120 Palaiseau, France

Email: vdhoeven@lix.polytechnique.fr

Surreal numbers form the ultimate extension of the field of real numbers with infinitely large and small quantities and in particular with all ordinal numbers. Hyperseries can be regarded as the ultimate formal device for representing regular growth rates at infinity. In this paper, we show that any surreal number can naturally be regarded as the value of a hyperseries at the first infinite ordinal . This yields a remarkable correspondence between two types of infinities: numbers and growth rates.

1.Introduction

1.1.Toward a unification of infinities

At the end of the 19-th century, two theories emerged for computations with infinitely large quantities. The first one was due to du Bois-Reymond [19, 20, 21], who developed a “calculus of infinities” to deal with the growth rates of functions in one real variable at infinity. The second theory of “ordinal numbers” was proposed by Cantor [13] as a way to count beyond the natural numbers and to describe the sizes of sets in his recently introduced set theory.

Du Bois-Reymond's original theory was partly informal and not to the taste of Cantor, who misunderstood it [25]. The theory was firmly grounded and further developed by Hausdorff and Hardy. Hausdorff formalized du Bois-Reymond's “orders of infinity” in Cantor's set-theoretic universe [24]. Hardy focused on the computational aspects and introduced the differential field of logarithmico-exponential functions [28, 29]: such a function is constructed from the real numbers and an indeterminate (that we think of as tending to infinity) using the field operations, exponentiation, and the logarithm. Subsequently, this led to the notion of a Hardy field [12].

As to Cantor's theory of ordinal numbers, Conway proposed a dramatic generalization in the 1970s. Originally motivated by game theory, he introduced the proper class of surreal numbers [14], which simultaneously contains the set of all real numbers and the class of all ordinals. This class comes with a natural ordering and arithmetic operations that turn into a non-Archimedean real closed field. In particular, , , , are all surreal numbers, where stands for the first infinite ordinal.

Conway's original definition of surreal numbers is somewhat informal and draws inspiration from both Dedekind cuts and von Neumann's construction of the ordinals:

“If and are any two sets of (surreal) numbers, and no member of is any member of , then there is a (surreal) number . All (surreal) numbers are constructed in this way.”

The notation is called Conway's bracket. Conway proposed to consider as the simplest number between and . In general, one may define a partial ordering on by setting (we say that is simpler than ) if there exist and with and . Conway's bracket is uniquely determined by this simplicity relation.

The ring operations on are defined in a recursive way that is both very concise and intuitive: given and , we define

It is quite amazing that these definitions coincide with the traditional definitions when and are real, but that they also work for the ordinal numbers and beyond. Subsequently, Gonshor also showed how to extend the real exponential function to [26] and this extension preserves all first order properties of [16]. Simpler accounts and definitions of can be found in [37, 9].

The theory of Hardy fields focuses on the study of growth properties of germs of actual real differentiable functions at infinity. An analogue formal theory arose after the introduction of transseries by Dahn and Göring [15] and, independently, by Écalle [22, 23]. Transseries are a natural generalization of the above definition of Hardy's logarithmico-exponential functions, by also allowing for infinite sums (modulo suitable precautions to ensure that such sums make sense). One example of a transseries is

In particular, any transseries can be written as a generalized series with real coefficients and whose (trans)monomials are exponentials of other (generally “simpler”) transseries. The support of such a series should be well based in the sense that it should be well ordered for the opposite ordering of the natural ordering on the group of transmonomials . The precise definition of a transseries depends on further technical requirements on the allowed supports. But for all reasonable choices, “the” resulting field of transseries possesses a lot of closure properties: it is ordered and closed under derivation, composition, integration, and functional inversion [22, 30, 18]; it also satisfies an intermediate value property for differential polynomials [32, 3].

It turns out that surreal numbers and transseries are similar in many respects: both and are real closed fields that are closed under exponentiation and taking logarithms of positive elements. Surreal numbers too can be represented uniquely as Hahn series with real coefficients and monomials in a suitable multiplicative subgroup of . Any transseries actually naturally induces a surreal number by substituting for and the map is injective [11].

But there are also differences. Most importantly, elements of can be regarded as functions that can be derived and composed. Conversely, the surreal numbers come equipped with the Conway bracket. In fact, it would be nice if any surreal number could naturally be regarded as the value of a unique transseries at . Indeed, this would allow us to transport the functional structure of to the surreal numbers. Conversely, we might equip the transseries with a Conway bracket and other exotic operations on the surreal numbers. The second author conjectured the existence of such a correspondence between and a suitably generalized field of the transseries [32, page 16]; see also [2] for a more recent account.

Now we already observed that at least some surreal numbers can be written uniquely as for some transseries . Which numbers and what kind of functions do we miss? Since a perfect correspondence would induce a Conway bracket on , it is instructive to consider subsets with and examine which natural growth orders might fit between and .

One obvious problem with ordinary transseries is that there exists no transseries that grows faster than any iterated exponential . Consequently, there exists no transseries with . A natural candidate for a function that grows faster than any iterated exponential is the first hyperexponential , which satisfies the functional equation

It was shown by Kneser [33] that this equation actually has a real analytic solution on . A natural hyperexponential on was constructed more recently in [8]. In particular, .

More generally, one can formally introduce the transfinite sequence of hyperexponentials of arbitrary strengths , together with the sequence of their functional inverses, called hyperlogarithms. Each with satisfies the equation

and there again exist real analytic solutions to this equation [38]. The function does not satisfy any natural functional equation, but we have the following infinite product formula for the derivative of every hyperlogarithm :

We showed in [6] how to define and for any and .

The traditional field of transseries is not closed under hyperexponentials and hyperlogarithms, but it is possible to define generalized fields of hyperseries that do enjoy this additional closure property. Hyperserial grow rates were studied from a formal point of view in [22, 23]. The first systematic construction of hyperserial fields of strength is due to Schmeling [38]. In this paper, we will rely on the more recent constructions from [17, 7] that are fully general. In particular, the surreal numbers form a hyperserial field in the sense of [7], when equipped with the hyperexponentials and hyperlogarithms from [6].

A less obvious problematic cut in the field of transseries arises by taking

Here again, there exists no transseries with . This cut has actually a natural origin, since any “tame” solution of the functional equation

(1.1)

lies in this cut. What is missing here is a suitable notion of “nested transseries” that encompasses expressions like

(1.2)

This type of cuts were first considered in [30, Section 2.7.1]. Subsequently, the second author and his former PhD student Schmeling developed an abstract notion of generalized fields of transseries [31, 38] that may contain nested transseries. However, it turns out that expressions like (1.2) are ambiguous: one may construct fields of transseries that contain arbitrarily large sets of pairwise distinct solutions to (1.1).

In order to investigate this ambiguity more closely, let us turn to the surreal numbers. The above cut induces a cut in . Nested transseries solutions to the functional equation (1.1) should then give rise to surreal numbers with and such that are all monomials in . In [5, Section 8], we showed that those numbers actually form a class that is naturally parameterized by a surreal number ( forms a so-called surreal substructure). Here we note that analogue results hold when replacing Gonshor's exponentiation by Conway's -map (which generalizes Cantor's -map when ). This was already noted by Conway himself [14, pages 34–36] and further worked out by Lemire [34, 35, 36]. Section 6 of the present paper will be devoted to generalizing the result from [5, Section 8] to nested hyperseries.

Besides the two above types of superexponential and nested cuts, no other examples of “cuts that cannot be filled” come naturally to our mind. This led the second author to conjecture [32, page 16] that there exists a field of suitably generalized hyperseries in such that each surreal number can uniquely be represented as the value of a hyperseries at . In order to prove this conjecture, quite some machinery has been developed since: a systematic theory of surreal substructures [5], sufficiently general notions of hyperserial fields [17, 7], and definitions of on the surreals that give the structure of a hyperserial field [8, 6].

Now one characteristic property of generalized hyperseries in should be that they can uniquely be described using suitable expressions that involve , real numbers, infinite summation, hyperlogarithms, hyperexponentials, and a way to disambiguate nested expansions. The main goal of this paper is to show that any surreal number can indeed be described uniquely by a hyperserial expression of this kind in . This essentially solves the conjecture from [32, page 16] by thinking of hyperseries in as surreal numbers in which we replaced by . Of course, it remains desirable to give a formal construction of that does not involve surreal numbers and to specify the precise kind of properties that our “suitably generalized” hyperseries should possess. We intend to address this issue in a forthcoming paper.

Other work in progress concerns the definition of a derivation and a composition on . Now Berarducci and Mantova showed how to define a derivation on that is compatible with infinite summation and exponentiation [10]. In [4, 1], it was shown that there actually exist many such derivations and that they all satisfy the same first order theory as the ordered differential field . However, as pointed out in [2], Berarducci and Mantova's derivation does not obey the chain rule with respect to . The hyperserial derivation that we propose to construct should not have this deficiency and therefore be a better candidate for the derivation on with respect to .

1.2.Outline of our results and contributions

In this paper, we will strongly rely on previous work from [5, 17, 7, 8, 6]. The main results from these previous papers will be recalled in Sections 2, 3, and 4. For the sake of this introduction, we start with a few brief reminders.

The field of logarithmic hyperseries was defined and studied in [17]. It is a field of Hahn series in the sense of [27] that is equipped with a logarithm , a derivation , and a composition . Moreover, for each ordinal , it contains an element such that

for all and all ordinals . Moreover, if the Cantor normal form of is given by with , then we have

The derivation and composition on satisfy the usual rules of calculus and in particular a formal version of Taylor series expansions.

In [7], Kaplan and the authors defined the concept of a hyperserial field to be a field of Hahn series with a logarithm and a composition law , such that various natural compatibility requirements are satisfied. For every ordinal , we then define the hyperlogarithm of strength by . We showed in [6] how to define bijective hyperlogarithms for which has the structure of a hyperserial field. For every ordinal , the functional inverse of is called the hyperexponential of strength .

The main aim of this paper is to show that any surreal number is not just an abstract hyperseries in the sense of [6], but that we can regard it as a hyperseries in . We will do this by constructing a suitable unambiguous description of in terms of , the real numbers, infinite summation, the hyperexponentials, and the hyperlogarithms.

If for some ordinary transseries , then the idea would be to expand as a linear combination of monomials, then to rewrite every monomial as an exponential of a transseries, and finally to recursively expand these new transseries. This process stops whenever we hit an iterated logarithm of .

In fact, this transserial expansion process works for any surreal number . However, besides the iterated logarithms (and exponentials) of , there exist other monomials such that is a monomial for all . Such monomials are said to be log-atomic. More generally, given , we say that is -atomic if for all . We write for the set of such numbers. If we wish to further expand an -atomic monomial as a hyperseries, then it is natural to pick such that is not -atomic, to recursively expand , and then to write .

Unfortunately, the above idea is slightly too simple to be useful. In order to expand monomials as hyperseries, we need something more technical. In Section 5, we show that every non-trivial monomial has a unique expansion of exactly one of the two following forms:

(1.3)

where , , and , with ; or

(1.4)

where , , with , , and where lies in . Moreover, if then it is imposed that , , and that cannot be written as where , , , and .

After expanding in the above way, we may pursue with the recursive expansions of and as hyperseries. Our next objective is to investigate the shape of the recursive expansions that arise by doing so. Indeed, already in the case of ordinary transseries, such recursive expansions may give rise to nested expansions like

(1.5)

One may wonder whether it is also possible to obtain expansions like

(1.6)

Expansions of the forms (1.5) and (1.6) are said to be well-nested and ill-nested, respectively. The axiom T4 for fields of transseries in [38] prohibits the existence of ill-nested expansions. It was shown in [10] that satisfies this axiom T4.

The definition of hyperserial fields in [6] does not contain a counterpart for the axiom T4. The main goal of section 4 is to generalize this property to hyperserial fields and prove the following theorem:

Theorem 1.1. Every surreal number is well-nested.

Now there exist surreal numbers for which the above recursive expansion process leads to a nested expansion of the form (1.5). In [5, Section 8], we proved that the class of such numbers actually forms a surreal substructure. This means that is isomorphic to for the restriction of to . In particular, although the nested expansion (1.5) is inherently ambiguous, elements in are naturally parameterized by surreal numbers in .

The main goal of Section 6 is to prove a hyperserial analogue of the result from [5, Section 8]. Now the expansion (1.5) can be described in terms of the sequence . More generally, in Section 6 we the define the notion of a nested sequence in order to describe arbitrary nested hyperserial expansions. Our main result is the following:

Theorem 1.2. Any nested sequence induces a surreal substructure of nested hyperseries.

In Section 7, we reach the main goal of this paper, which is to uniquely describe any surreal number as a generalized hyperseries in . This goal can be split up into two tasks. First of all, we need to specify the hyperserial expansion process that we informally described above and show that it indeed leads to a hyperserial expansion in , for any surreal number. This will be done in Section 7.2, where we will use labeled trees in order to represent hyperserial expansions. Secondly, these trees may contain infinite branches (also called paths) that correspond to nested numbers in the sense of Section 6. By Theorem 1.2, any such nested number can uniquely be identified using a surreal parameter. By associating a surreal number to each infinite branch, this allows us to construct a unique hyperserial description in for any surreal number and prove our main result:

Theorem 1.3. Every surreal number has a unique hyperserial description. Two numbers with the same hyperserial description are equal.

2.Ordered fields of well-based series

2.1.Well-based series

Let be a totally ordered (and possibly class-sized) abelian group. We say that is well-based if it contains no infinite ascending chain (equivalently, this means that is well-ordered for the opposite ordering). We denote by the class of functions whose support

is a well-based. The elements of are called monomials and the elements in are called terms. We also define

and elements are called terms in .

We see elements of as formal well-based series where for all . If , then is called the dominant monomial of . For , we define and . For , we sometimes write if . We say that a series is a truncation of and we write if . The relation is a well-founded partial order on with minimum .

By [27], the class is field for the pointwise sum

and the Cauchy product

where each sum is finite. The class is actually an ordered field, whose positive cone is defined by

The ordered group is naturally embedded into .

The relations and on extend to by

We also write whenever and . If are non-zero, then (resp. , resp. ) if and only if (resp. , resp. ).

We finally define

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

2.2.Well-based families

If is a family in , then we say that is well-based if

  1. is well-based, and

  2. is finite for all .

Then we may define the sum of as the series

If is another field of well-based series and is -linear, then we say that is strongly linear if for every well-based family in , the family in is well-based, with

2.3.Logarithmic hyperseries

The field of logarithmic hyperseries plays an important role in the theory of hyperseries. Let us briefly recall its definition and its most prominent properties from [17].

Let be an ordinal. For each , we introduce the formal hyperlogarithm and define to be the group of formal power products with . This group comes with a monomial ordering that is defined by

By what precedes, is an ordered field of well-based series. If are ordinals with , then we define to be the subgroup of of monomials with whenever . As in [17], we define

We have natural inclusions , hence natural inclusions .

The field is equipped with a derivation which satisfies the Leibniz rule and which is strongly linear: for each , we first define the logarithmic derivative of by . The derivative of a logarithmic hypermonomial is next defined by

Finally, this definition extends to by strong linearity. Note that for all . For and , we will sometimes write .

Assume that for a certain ordinal . Then the field is also equipped with a composition that satisfies:

The same properties hold for the composition , when is replaced by . For , the map is injective, with range [17, Lemma 5.11]. For , we define to be the unique series in with .

3.Surreal numbers as a hyperserial field

3.1.Surreal numbers

Following [26], we define as the class of sequences

of “signs” indexed by arbitrary ordinals . We will write for the domain of such a sequence and for its value at . Given sign sequences and , we define

Conway showed how to define an ordering, an addition, and a multiplication on that give the structure of a real closed field [14]. See [5, Section 2] for more details about the interaction between and the ordered field structure of . By [14, Theorem 21], there is a natural isomorphism between and the ordered field of well-based series , where is a certain subgroup of . We will identify those two fields and thus regard as a field of well-based series with monomials in .

The partial order contains an isomorphic copy of obtained by identifying each ordinal with the constant sequence of length . We will write to specify that is either an ordinal or the class of ordinals. The ordinal , seen as a surreal number, is the simplest element, or -minimum, of the class .

For , we write and for the non-commutative ordinal sum and product of and , as defined by Cantor. The surreal sum and product and coincide with the commutative Hessenberg sum and product of ordinals. In general, we therefore have and .

For , we write for the ordinal exponentiation of base at . Gonshor also defined an exponential function on with range . One should not confuse with , which yields a different number, in general. We define

Recall that every ordinal has a unique Cantor normal form

where , and with . The ordinals are called the exponents of and the integers its coefficients. We write (resp. ) if each exponent of the Cantor normal form of satisfies (resp. ).

If are ordinals, then we write if , we write if there exists an with , and we write if both and hold. The relation is a quasi-order on . For with and , we have . In particular, we have for all .

If is a successor, then we define to be the unique ordinal with . We also define if is a limit. Similarly, if , then we write .

3.2.Hyperserial structure on

We already noted that Gonshor constructed an exponential and a logarithm on and , respectively. We defined hyperexponential and hyperlogarithmic functions of all strengths on in [6]. In fact, we showed [6, Theorem 1.1] how to construct a composition law with the following properties:

For , and , we have and

For , the function is a strongly linear field morphism.

For , and with , we have

For and with , we have .

For and , there is with .

Note that the composition law on also satisfies to (but not ), with each occurrence of being replaced by .

3.3.Hyperlogarithms

For , we write for the function , called the hyperlogarithm of strength . By and , this is a strictly increasing bijection. We sometimes write for . We write for the functional inverse of , called the hyperexponential of strength .

For with , the relation in , combined with , yields

(3.1)

For , the relation in , combined with , yields

(3.2)

and we call this relation the functional equation for .

Let and write for the coefficient in in the Hahn series representation of . There is a unique infinitesimal number with . We write for the natural logarithm on . The function defined by

(3.3)

is called the logarithm on . This is a strictly increasing morphism which extends . It also coincides with the logarithm on that was defined by Gonshor.

3.4.Atomicity

Given , we write for the class of numbers with for all . Those numbers are said to be -atomic and they play an important role in this paper. Note that and

for all , in view of (3.1). There is a unique -atomic number [6, Proposition 6.20], which is the simplest positive infinite number .

Each hyperlogarithmic function with is essentially determined by its restriction to , through a generalization of (3.3). More precisely, for , there exist and with . Moreover, the family is well-based, and the hyperlogarithm is given by

(3.4)

3.5.Hyperexponentiation

Definition 3.1. [7, Definition 6.10] We say that is -truncated if , i.e. if is positive and purely large. For , we say that is -truncated if

If is defined, then is -truncated if and only if , for all .

Given with , we write for the class of -truncated numbers. Note that . We will sometimes write when . For , there is a unique -maximal truncation of which is -truncated. By [7, Proposition 7.17], the classes

(3.5)

with form a partition of into convex subclasses. Moreover, the series is both the unique -truncated element and the -minimum of the convex class containing . We have

by [6, Proposition 7.6]. This allows us to define a map by . In other words,

for all (see also [7, Corollary 7.23]).

The formulas (3.3) and (3.4) admit hyperexponential analogues. For all , there is a with . For any such , there is a family with such that is well-based and

(3.6)

See [7, Section 7.1] for more details on . The number is a monomial with , so

(3.7)

4.Surreal substructures

In [5], we introduced the notion of surreal substructure. A surreal substructure is a subclass of such that and are isomorphic. The isomorphism is unique and denoted by . For the study of as a hyperserial field, many important subclasses of turn out to be surreal substructures. In particular, given , it is known that the following classes are surreal substructures:

We will prove in Section 6 that certain classes of nested numbers also form surreal substructures.

4.1.Cuts

Given a subclass of and , we define

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 [5, 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 [5, Proposition 4.11(b)].

4.2.Cut equations

Let be a subclass, let be a surreal substructure and be a function. Let be functions defined for cut representations in 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

for all cut representations in . For instance, given , consider the translation on . By [26, Theorem 3.2], we have the following uniform cut equation for on :

(4.1)

Let with and set . We have the following uniform cut equations for on and on [6, Section 8.1]:

(4.2)
(4.3)
(4.4)
(4.5)

where

4.3.Function groups

A function group on a surreal substructure is a set-sized group of strictly increasing bijections under functional composition. We see elements of as actions on and sometimes write and for rather than and . We also write for the functional inverse of .

Given such a function group , the collection of classes

with forms a partition of into convex subclasses. For subclasses , we write . An element is said to be -simple if it is the simplest element inside . We write for the class of -simple elements. Given , we also define to be the unique -simple element of . The function is a non-decreasing projection of onto . The main purpose of function groups is to define surreal substructures:

Proposition 4.1. [5, Theorem 6.7 and Proposition 6.8] The class is a surreal substructure. We have the uniform cut equation

(4.6)

Note that for , we have if and only if . We have the following criterion to identify the -simple elements inside .

Proposition 4.2. [5, Lemma 6.5] An element of is -simple if and only if there is a cut representation of in with . Equivalently, the number is -simple if and only if .

Given be sets of strictly increasing bijections , we define

If , then we say that is pointwise cofinal with respect to . For , we also write or instead of and .

Given a function group on , we define a partial order on by . We will frequently rely on the elementary fact that this ordering is compatible with the group structure in the sense that

Given a set of strictly increasing bijections , we define to be the smallest function group on that is generated by , i.e. .

4.4.Remarkable function groups

The examples of surreal substructures from the beginning of this section can all be obtained as classes of -simplest elements for suitable function groups that act on , , or , as we will describe now. Given and , we first define

For , we then have the following function groups

Now the action of on yields the surreal substructure as class of -simplest elements. All examples from the beginning of this section can be obtained in a similar way:

We have

Let and . We will need a few inequalities from [6]. The first one is immediate by definition and the fact that . The others are [6, Lemma 6.9, Lemma 6.11, and Proposition 6.17], in that order:

(4.7)
(4.8)
(4.9)
(4.10)

From (4.10), we also deduce that

(4.11)

5.Well-nestedness

In this section, we prove Theorem 1.1, i.e. that each number is well-nested. In Section 5.1 we start with the definition and study of hyperserial expansions. We pursue with the study of paths and well-nestedness in Section 5.2.

The general idea behind our proof of Theorem 1.1 is as follows. Assume for contradiction that there exists a number that is not well-nested and choose a simplest (i.e. -minimal) such number. By definition, contains a so-called “bad path”. For the ill-nested number from (1.6), that would be the sequence

From this sequence, we next construct a “simpler” number like

that still contains a bad path

thereby contradicting the minimality assumption on . In order to make this idea work, we first need a series of “deconstruction lemmas” that allow us to affirm that is indeed simpler than ; these lemmas will be listed in Section 5.3. We will also need a generalization of the relation that was used by Berarducci and Mantova to prove the well-nestedness of as a field of transseries; this will be the subject of Section 5.4. We prove Theorem 1.1 in Section 5.5. Unfortunately, the relation does not have all the nice properties of . For this reason, Sections 5.4 and 5.5 are quite technical.

5.1.Hyperserial expansions

Recall that any number can be written as a well-based series. In order to represent numbers as hyperseries, it therefore suffices to devise a means to represent the infinitely large monomials in . We do this by taking a hyperlogarithm of the monomial and then recursively applying the same procedure for the monomials in this new series. This procedure stops when we encounter a monomial in .

Technically speaking, instead of directly applying a hyperlogarithm to the monomial, it turns out to be necessary to first decompose as a product and write as a hyperexponential (or more generally as the hyperlogarithm of a hyperexponential). This naturally leads to the introduction of hyperserial expansions of monomials , as we will detail now.

Definition 5.1. We say that a purely infinite number is tail-atomic if , for certain , , and .

Definition 5.2. Let . Assume that there are , , , and such that

(5.1)

with . Then we say that (5.1) is a hyperserial expansion of type I if

We say that (5.1) is a hyperserial expansion of type II if and , so that and

(5.2)

Formally speaking, hyperserial expansions can be represented by tuples . By convention, we also consider

to be a hyperserial expansion of the monomial ; this expansion is represented by the tuple .

Example 5.3. We will give a hyperserial expansion for the monomial

and show how it can be expressed as a hyperseries. Note that

is tail-atomic since is log-atomic. Now is a hyperserial expansion of type II and we have . Hence is a hyperserial expansion.

Let , so . We may further expand each monomial in . We clearly have . We claim that . Indeed, if we could write for some and , then and would both be monomials, which cannot be. Note that , so is a hyperserial expansion of type I. We also have where is tail expanded. Thus is a hyperserial expansion. Note finally that is a hyperserial expansion. We thus have the following “recursive” expansion of :

(5.3)

Lemma 5.4. Any has a hyperserial expansion.

Proof. We first prove the result for , by induction with respect to the simplicity relation . The -minimal element of is , which satisfies (5.2) for and . Consider such that the result holds on . By [6, Proposition 6.20], the monomial is not -atomic. So there is a maximal with , and we have by our hypothesis.

If there is no ordinal such that , then we have . So setting , and , we are done. Otherwise, let be such that . We cannot have by definition of . So there is a unique ordinal and a unique natural number such that and . Note that . We must have : otherwise, where and are monomials. We deduce that and . Note that , , and , so . We deduce that . The induction hypothesis yields a hyperserial expansion . Since is log-atomic, we must have and . If , then , since . Thus is a hyperexponential expansion of type II. If , then likewise and thus is a hyperexponential expansion of type I. This completes the inductive proof.

Now let and set . If is tail-atomic, then there are , and with . Applying the previous arguments to , we obtain elements with and , or an ordinal with . Then or is a hyperserial expansion. If is not tail-atomic, then we have is a hyperserial expansion of type I.

Lemma 5.5. Consider a hyperserial expansion . Let and define if is a successor ordinal and if is a limit ordinal. Let

  1. Then is -atomic if and only if and either or .

  2. If , then .

Proof. We first prove ). Assume that is -atomic. Assume for contradiction that and let denote the least non-zero term in the Cantor normal form of . Since , we have so is a monomial. But where is a monomial: a contradiction. So . If then we are done. Otherwise , so we must have , whence . Conversely, assume that or , and that . If , then then for all , we have where , so is a monomial, whence . If , then for all , we have , whence . This proves ).

Now assume that . So is -atomic by ). If then we conclude that . If , then let denote the least non-zero term in its Cantor normal form. We have and , so .

Corollary 5.6. Let , , , and . If , then is a successor ordinal and for some .

Proof. Since , we must have . By Lemma 5.4, we have a hyperserial expansion . Since is log-atomic, we have , whence and . So . We have so by Lemma 5.5(a), we have . It follows that is a hyperserial expansion. But then and Lemma 5.5(a) imply that . The condition that now gives , whence is a successor and for a certain , as claimed.

Lemma 5.7. Any has a unique hyperserial expansion (that we will call the hyperserial expansion, henceforth).

Proof. Consider a monomial with

where , , , , , and . Assume for contradiction that we can write as a hyperserial expansion of type I with . Note in particular that , so is log-atomic. We have

If , then , , , and is not tail-atomic. But , where , so is tail-atomic: a contradiction. Hence . Note that and are both the least term of . It follows that , , and

(5.4)

Since , we have

Now , so and thus . In particular . Taking hyperexponentials on both sides of (5.4), we may assume without loss of generality that or that the least exponents and in the Cantor normal forms of resp. differ. If , then we decompose where and . Since , applying Lemma 5.5(a) twice (for and ) gives and , whence . But then , where by Lemma 5.5(a). So : a contradiction. Assume now that . Lemma 5.5(a) yields both and , which contradicts (5.4).

Taking and , this proves that no two hyperserial expansions of distinct types I and II can be equal. Taking with , this proves that no two hyperserial expansions of type I with can be equal.

The two remaining cases are hyperserial expansions of type II and hyperserial expansions and of type I with . Consider a monomial with the hyperserial expansions of type II. As above we have , , and . We deduce that , so the expansions coincide.

Finally, consider a monomial with two hyperserial expansions of type I

(5.5)

If , then we have and and , whence , so we are done.

Assume now that . Taking logarithms in (5.5), we see that , , and

(5.6)

We may assume without loss of generality that . Assume for contradiction that . Taking hyperexponentials on both sides of (5.6), we may assume without loss of generality that or that the least exponents and in the Cantor normal forms of resp. differ. On the one hand, Lemma 5.5(a) yields . Note in particular that , since . On the other hand, if , then Lemma 5.5(a) yields ; if , then . Thus (5.6) is absurd: a contradiction. We conclude that . Finally yields , so the expansions are identical.

Lemma 5.8. If is a hyperserial expansion of type I, then we have

Proof. Assume for contradiction that . In particular . Since , there is with , so : a contradiction.

5.2.Paths and subpaths

Let be an ordinal with and note that for all . Consider a sequence

We say that is a path if there exist sequences , , , , and with

We call the length of and we write . We say that is infinite if and finite otherwise. We set . For , we define

By Lemma 5.8, those cases are mutually exclusive so is well-defined. For , we say that is a path in if .

For , we let denote the path of length in with

Example 5.9. Let us find all the paths in the monomial of Example 5.3. We have a representation (5.3) of as a hyperseries

which by Lemma 5.7 is unique. There are nine paths in , namely

Note that the paths which cannot be extended into strictly longer paths are those whose last value is a real number or .

Infinite paths occur in so-called nested numbers that will be studied in more detail in Section 6.

Definition 5.10. Let and let be a path in a. We say that an index is bad for if one of the following conditions is satisfied

  1. is not the -minimum of ;

  2. and ;

  3. and and ;

  4. and and and .

The index is good for if it is not bad for .

If is infinite, then we say that it is good if is good for all but a finite number of indices. In the opposite case, we say that is a bad path. An element is said to be well-nested every path in is good.

Remark 5.11. The above definition extends the former definitions of paths in [30, 38, 10]. More precisely, a path with with (whence ) for all , corresponds to a path for these former definitions. The validity of the axiom T4 for means that those paths are good. With Theorem 1.1, we will extend this result to all paths.

Lemma 5.12. For and for any path in , we have . For and for any path in , we have .

Proof. Let and let be a path in . If there is an ordinal with , then the hyperserial expansion of is , so if and otherwise. If there is an ordinal with , then the hyperserial expansion of is and .

Assume now that . If is not tail-atomic, then hyperserial expansion of is . If is tail-atomic, then the hyperserial expansion of is for a certain log-atomic . In both cases, is a path in some monomial in , whence and , by the previous argument.

Definition 5.13. Let be paths. We say that is a subpath of , or equivalently that extends , if there exists a with . For , we say that is a subpath in if there is a path in such that is a subpath of . We say that shares a subpath with if there is a subpath of which is a subpath in .

Let be a finite path and let be a path with . Then we define to be the path of length .

Lemma 5.14. Let and . Let be a path in with . Then is a subpath in .

Proof. By Lemma 5.12, we have . If has a hyperserial expansion of the form , then must be a path in . So is non-zero and thus . It follows that is a path in . Otherwise, let be the hyperserial expansion of . If is a path in , then it is a path in as above. Otherwise, it is a path in . Assume that . If , then we have and so is a path in . If , then where is a hyperserial expansion, so is a path in . Assume now that , so , , and . We must have so there are and with and . We have where is a hyperserial expansion, so is a path in .

Lemma 5.15. Let , and . If is a path in with , then is a subpath in .

Proof. We prove this by induction on , for any number . We consider , and a fixed path in with .

Assume that . We have and . Assume that for certain , , and . Let be the hyperserial expansion of . If , then and the hyperserial expansion of is . Therefore is a subpath in . If , then the hyperserial expansion of is . Therefore is a subpath in . Finally, if is not tail-atomic, then is a subpath in , where is the sign of .

Now assume that , , and that the result holds strictly below . We have where is a subpath in by the previous argument. We have for a certain , so is a path in . The induction hypothesis on implies that is a subpath in .

Assume now that and that the result holds strictly below . Write . By (3.7), there exist , , and with and

Assume for contradiction that there is a with . We must have , so there are a number and an ordinal with . We have . By Lemma 5.12, this contradicts the fact that . So by Lemma 5.4, there exist and with , , , and . Since , we must have so there are a number and an ordinal with (note that whenever ). Thus is a monomial with hyperserial expansion . There is no path in of length , so must be a path in . We deduce that is a path in . Consequently, is a path in with . Applying times Lemma 5.14, we deduce that is a subpath in , hence in . Consider a path in with for a certain . Applying the induction hypothesis for and in the roles of and , the path is a subpath in . Therefore is a subpath in . We deduce as in the case that is a subpath in .

Lemma 5.16. Let , and with . Let be a path in with . Then is a subpath in .

Proof. Let be a hyperserial expansion. The condition implies , whence is also a hyperserial expansion. In particular is a subpath in .

Corollary 5.17. Let , with , and . If is an infinite path, then shares a subpath with if and only if it shares a subpath with .

Proof. Write in Cantor normal form, with and and let

for all .

Assume that shares a subpath with . In other words, there is a path in which has a common subpath with . The path must be infinite, so by Lemma 5.15, it shares a subpath with . Let us prove by induction on that shares a subpath with . Assuming that this holds for , we note that is -atomic, hence -atomic. So shares a subpath with by Lemma 5.14 and the induction hypothesis. We conclude by induction that shares a subpath with .

Suppose conversely that shares a subpath with . By induction on , it follows from Lemma 5.15 that shares a subpath with . Applying Lemma 5.14 to , we conclude that shares a subpath with .

5.3.Deconstruction lemmas

In this subsection, we list several results on the interaction between the simplicity relation and various operations in .

Lemma 5.18. [26, Theorem 3.3] For , we have

Lemma 5.19. [26, Theorem 5.12(a)] For and , we have

Lemma 5.20. [5, Proposition 4.20] Let . For with , we have

Lemma 5.21. [10, Corollary 4.21] For , we have

Lemma 5.22. [10, Proposition 4.23] Given in with , we have

Lemma 5.23. [10, Proposition 4.24] Given with , we have

Lemma 5.24. Let and , let with , and let with . Then

Proof. The condition yields . We have by Lemma 5.23. The identity implies that , whence by Lemma 5.21. Consequently, , by Lemma 5.22. Since , we may apply Lemma 5.22 to and to obtain . We conclude using the transitivity of .

Lemma 5.25. Let with . For with , we have

Proof. By (4.5), we have

Since , we have and , whence

Furthermore, we have , so . We conclude that .

5.4.Nested truncation

In [10, Section 8], the authors prove the well-nestedness axiom for by relying on a well-founded partial order that is defined by induction. This relation has the additional property that

In this subsection, we define a similar relation on that will be instrumental in deriving results on the structure of . However, this relation does not satisfy for all .

Given , we define

where is a sequence of relations that are defined by induction on , as follows. For , we set , if or if there exist decompositions

with and . Assuming that has been defined, we set if we are in one of the two following configurations:

Configuration I

We may decompose and as

(5.7)
(5.8)

where , , , , , ,

and . If , then we also require that .

Configuration II

We may decompose and as

(5.9)
(5.10)

where , , , , , and .

Warning 5.26. Taking in the first configuration, we see that extends . However, the relation is neither transitive nor anti-symmetric. Furthermore, as we already noted above, we do not have .

Lemma 5.27. Let . Let be numbers of the form

where , with , and . If for sufficiently large , then

Proof. Let and . Assume for contradiction that and . Assume first that , so . Then . Let be such that . Since , we deduce that , whence . Modulo replacing by , it follow that we may assume without loss of generality that for some and some monomial .

On the one hand, is not -truncated, so there are and with and . We may choose for certain and , so . On the other hand, is -truncated, so we have

We deduce that . If is a successor, then choosing , we obtain , so : a contradiction. Otherwise, by [7, (2.4)], where . Thus , whence : a contradiction.

We now treat the general case. By a similar argument as above, we may assume without loss of generality that . Assume that . Since is not -truncated, there exists a with , whence . But is -truncated, so . In particular , so our hypothesis implies that : a contradiction.

Assume now that . As in the first part of the proof, there are and with and . Recall that for sufficiently large . Take and such that

(5.11)

Then . If is a successor, then choosing yields , which contradicts the fact that and are infinitesimal. So is a limit. Writing , we have . As in the first part of the proof, we obtain , so . In view of (5.11), we also obtain , so : a contradiction.

Lemma 5.28. Let with . For , we have

Proof. Assume that . Let and let be its functional inverse in . We have by (4.10, 4.11), whence . Furthermore, , so

(5.12)

We want to prove that . By (5.12), it is enough to prove that there is a such that the inequality holds on .

Assume that . Setting , we have , whence , and .

Assume that . We have so by (4.8). Thus . Consequently, , as claimed.

If are numbers, then we write for the interval .

Proposition 5.29. For with and , any infinite path in shares a subpath with .

Proof. We prove this by induction on with . Let be an infinite path in . Assume that . If , then we have so is a path in . Otherwise, there are , and with and . Then for certain , and with . We must have . If is a path in , then it is a path in . Otherwise, it is a path in , so is a subpath in , hence in .

We now assume that where and that the result holds for all and with and . Assume first that is in Configuration I, and write

Then we can write like in the case when . If is a path in , then it is a path in . So we may assume that is a path in . Note that we have . Setting , we observe that , whence is the hyperserial expansion of . If is a path in , then it is a path in .

Suppose that is not a path in . Assume first that , so , , and is a path in . Then Lemma 5.14 implies that is a subpath in , so is a subpath in . Otherwise, consider the hyperserial expansion , of . Since is not a path in , it must be a path in . The number is -atomic, so we must have and . There are and such that . Therefore . It follows by Corollary 5.17 that shares a subpath with , whence so does .

Let . Recall that , so . Now (4.8) implies that , so . The function is non-decreasing, so . But , so the induction hypothesis yields that , and thus , shares a subpath with . We deduce with Lemma 5.15 that shares a subpath with , hence with .

Assume now that is in Configuration II, and write

Note that we also have . We may again assume that is a path in . Write , where , , and . Then where . We deduce by induction that shares a subpath with . By Lemma 5.15, it follows that shares a subpath with , hence with . This concludes the proof.

Lemma 5.30. Let and with . Let be of the form

with , , , , , and . Consider an infinite path in with .

  1. If , then shares a subpath with .

  2. If and , then shares a subpath with .

  3. If and and , then shares a subpath with .

Proof. i. If , then we have , so . Let with . Since and are monomials, we have , whence . Our assumption that also implies . Hence . Now shares a subpath with , by Lemma 5.15. Since , Proposition 5.29 next implies that shares a subpath with . Using Lemma 5.14, we conclude that shares a subpath with , and hence with .

ii. Let with . It is enough to prove that shares a subpath with . Since , , and are monomials, we have . Let , so that . In particular, we have . Moreover , so using Lemma 5.15 and Proposition 5.29, we deduce in the same way as above that shares a subpath with . If , then is the hyperserial expansion of , so shares a subpath with . If , then the hyperserial expansion of must be of the form , since otherwise would have at least two elements in its support. We deduce that shares a subpath with and that the hyperserial expansion of is . Therefore shares a subpath with .

iii. We assume that is not -truncated whereas and . If , then we must have , which means that or that and . But then : a contradiction.

Assume that . By Lemma 5.27, we may assume without loss of generality that . The assumption on and the fact that imply that is non-zero. Write

So and . Note that must be infinitesimal since is not -truncated. Thus is also infinitesimal. By Lemma 5.27, we deduce that . We have , so , since and are both -truncated. Since is not -truncated, there is an ordinal with . If , then , because is -truncated. Thus . If , then , because and are -truncated. Now , since . We again deduce that .

In both cases, we have where , so shares a subpath with , by Proposition 5.29. It follows by Corollary 5.17 that shares a subpath with .

5.5.Well-nestedness

We now prove that every number is well-nested. Throughout this subsection, will be an infinite path inside a number . At the beginning of Section 5.2 we have shown how to attach sequences , , etc. to this path. In order to alleviate notations, we will abbreviate , , , , , , and for all .

We start with a technical lemma that will be used to show that the existence of a bad path in implies the existence of a bad path in a strictly simpler number than .

Lemma 5.31. Let , let be an infinite path in and let such that every index is good for . For , let , , and , so that and

(k<i)

Let and let be a number with and

(5.13)

for a certain with , and whenever . For , we define

(5.14)

Assume that shares a subpath with . If shares no subpath with any of the numbers , then we have , and shares a subpath with .

Proof. Using backward induction on , let us prove for that

(5.15)k
(5.16)k
(5.17)k
(5.18)k
(5.20)k
(5.21)k

and that (5.19) and (5.21) also hold for .

We first treat the case when . Note that since it contains a subpath, so or . From our assumption that and the fact that if , we deduce that . Hence and (5.15). Note that (5.19) and (5.20) follow immediately from the other assumptions on . If then . If , then , since and . Hence by Lemma 5.21 and by Lemmas 5.19 and 5.22. Finally, by Lemma 5.20, so (5.21) holds in general. Recall that is a subpath in , but that it shares no subpath with or . In view of (5.20), we deduce (5.16) from Lemma 5.30(i) and (5.17) from Lemma 5.30(ii). Combining (5.16), (5.17) and (5.20) with the relation , we finally obtain (5.18).

Let and assume that (5.155.21) hold for all . We shall prove (5.155.21) if , as well as (5.19) and (5.21). Recall that

(5.15)

Recall that . If or , then and (5.165.17) imply (5.15). Assume now that . It follows since that , so and . Since is a hyperserial expansion, we must have , so . The result now follows from (5.15) and Lemma 5.28.

(5.19)

We know by (5.19) that shares a subpath with . Since , we deduce with Corollary 5.17 that also shares a subpath with , hence with . In view of (5.16) and Lemma 5.16, we see that shares a subpath with . Hence (5.17) gives that shares a subpath with .

(5.16)

By (5.18), we have . Now shares a subpath with by (5.19), but it shares no subpath with . Lemma 5.30(i) therefore yields the desired result .

(5.17)

As above, shares a subpath with , but no subpath with . We also have and , so (5.17) follows from Lemma 5.30(ii).

(5.18)

We obtain (5.18) by combining (5.155.18) and (5.20).

(5.20)

The path shares a subpath with , but no subpath with . By what precedes, we also have and . Note finally that . Hence , by applying Lemma 5.30(iii) with , , , , and in the roles of , , , , and .

(5.21)

It suffices to prove that , since

(by Lemma 5.21)
(by Lemma 5.22)
(by Lemma 5.20)

Assume that and recall that

By Lemma 5.25, it suffices to prove that and that for all . The first relation holds by (5.21). By (5.15), we have . Therefore by Lemma 5.25. This yields the result.

Assume now that . For , let

We will prove, by a second descending induction on , that the monomials and satisfy the premises of Lemma 5.24, i.e. , , and . It will then follow by Lemma 5.24 that , thus concluding the proof.

If , then , because . In particular . Moreover, follows from our assumption that , the fact that , and Lemmas 5.22 and 5.21. If , then we have because . Otherwise, we have .

Now assume that , that the result holds for , and that . Again implies that . The relation and Lemmas 5.18, 5.19, and 5.20 imply that . If , then by (5.16). Otherwise, we have , because . Since , the number is not tail-atomic, so we must have . This entails that and . By the induction hypothesis at , we have . We deduce that , so

We deduce by induction that (5.21) is valid.

This concludes our inductive proof. The lemma follows from (5.21) and (5.19).

We are now in a position to prove our first main theorem.

Proof of Theorem 1.1. Assume for contradiction that the theorem is false. Let be a -minimal ill-nested number and let be a bad path in . Let be the smallest bad index in . As in Lemma 5.31, we define , , and for all . We may assume that , otherwise the number is ill-nested and satisfies : a contradiction.

Assume for contradiction that there is a such that or is ill-nested. Set if is ill-nested and otherwise. If , then cannot share a subpath with , so by Lemma 5.30, and is ill-nested. In general, it follows that is ill-nested. Let be a bad path in and set . Then we may apply Lemma 5.31 to , , and in the roles of , , and . Since , this yields an ill-nested number : a contradiction.

Therefore the numbers are well-nested. Since is bad for , one of the four cases listed in Definition 5.10 must occur. We set

By construction, we have . Furthermore shares a subpath with , so there exists a bad path in . We have by Lemma 5.27. If Definition 5.10(4) occurs, then we must have so is written as in (5.13) with in the role of and . Otherwise, is as in (5.13) for . Setting , it follows that we may apply Lemma 5.31 to and in the roles of and . We conclude that there exists an ill-nested number : a contradiction.

6.Surreal substructures of nested numbers

In the previous section, we have examined the nature of infinite paths in surreal numbers and shown that they are ultimately “well-behaved”. In this section, we work in the opposite direction and show how to construct surreal numbers that contain infinite paths of a specified kind. We follow the same method as in [5, Section 8].

Let us briefly outline the main ideas. Our aim is to construct “nested numbers” that correspond to nested expressions like

(6.1)

Nested expressions of this kind will be presented through so-called coding sequences . Once we have fixed such a coding sequence , numbers of the form (6.1) need to satisfy a sequence of natural inequalities: for any with , we require that

Numbers that satisfy these constraints are said to be admissible. Under suitable conditions, the class Ad of admissible numbers forms a convex surreal substructure. This will be detailed in Sections 6.1 and 6.2, where we will also introduce suitable coordinates

for working with numbers in .

The notation (6.1) also suggests that each of the numbers , , should be a monomial. An admissible number is said to be nested if this is indeed the case. The main result of this section is Theorem 1.2, i.e. that the class of nested numbers forms a surreal substructure. In other words, the notation (6.1) is ambiguous, but can be disambiguated using a single surreal parameter.

6.1.Coding sequences

Definition 6.1. Let . We say that is a coding sequence if for all , we have

  1. ;

  2. ;

  3. ;

  4. ;

  5. .

Taking for all , we obtain a reformulation of the notion of coding sequences in [5, Section 8.1]. If is a coding sequence and , then we write

which is also a coding sequence.

Lemma 6.2. Let be an infinite path in a number without any bad index for . Let and for all . Then is a coding sequence.

Proof. Let . We have because is a good index for . We have and by the definition of hyperserial expansions. If and , then we have because by the definition of paths. Lemma 5.27 also yields . This proves the conditions ) and ) for coding sequences. Assume that . Then by the definition of hyperserial expansions, we have and is not tail-atomic. Assume that . Then so . We have where and is not tail-atomic. This implies that is not log-atomic, so . Thus ) is valid.

Assume that . Recall that , so . Since , we have , whence . This proves ).

Assume now for contradiction that there is an with for all . By ), we have for all , and the sequence is non-increasing, hence eventually constant. Let with for all . For , we have so . Therefore is -atomic: a contradiction. We deduce that ) holds as well.

We next fix some notations. For all with , we define partial functions , and on by

The domains of these functions are assumed to be largest for which these expressions make sense. We also write

We note that on their respective domains, the functions , , and are strictly increasing if , , and , respectively, and strictly decreasing in the contrary cases. We will write and for the partial inverses of and . We will also use the abbreviations

For all , we set

Note that

The following lemma generalizes [5, Lemma 8.1].

Lemma 6.3. If , then is well defined for all .

Proof. Let us prove the lemma by induction on . The result clearly holds for . Assuming that is well defined, let be minimal such that or . Note that we have , so where . Applying to the inequality

we obtain

Now if , then

whence

Both in the cases when and when , it follows that is bounded from below by the hyperexponential of a number. Thus is well defined and so is each for . If , then we have and

Hence

Both in the cases when and when , it follows that is bounded from below by the hyperexponential of a number, so is well defined and so is each for .

6.2.Admissible sequences

Definition 6.4. Let be a coding sequence and let . We say that is -admissible if is well defined for all and

We say that is admissible if there exists a -admissible number.

Note that we do not ask that be a hyperserial expansion, nor even that be a monomial. For the rest of the section, we fix a coding sequence . We write for the class of -admissible numbers. If , then the definition of implicitly assumes that is well defined for all . Note that if is admissible, then so is for . We denote by the corresponding class of -admissible numbers.

The main result of this subsection is the following generalization of [5, Proposition 8.2]:

Proposition 6.5. We have .

Proof. Let and let . We have . If , then is strictly increasing so we have

If , then is strictly decreasing and likewise we obtain .

We have . If , then is strictly increasing so we have

Likewise, we have if .

Assume that and . If , then we have . Hence

If , then we have , whence

Symmetric arguments apply when and .

We deduce by definition of that .

As a consequence of this last proposition and [5, Proposition 4.29(a)], the class is a surreal substructure if and only if is admissible.

Example 6.6. Consider the coding sequence where for all , we have

We use the notations from Section 6.1. We claim that is admissible. Indeed for , set

Given and , we have and . We deduce that , whence is admissible.

Lemma 6.7. Let and be such that and have the same sign and the same dominant monomial. Then .

Proof. For , we write if and and have the same sign. Let us prove by induction on that is defined and that . Since this implies that , that , and that if , this will yield .

The result follows from our hypothesis if . Assume now that and let us prove that . Let

We have , so is defined. Moreover so . Since , we deduce that , whence in particular . This concludes the proof.

Corollary 6.8. We have .

Proof. For , and , we have so . We conclude with the previous lemma.

Lemma 6.9. For and , we have .

Proof. Let be minimal with or . We thus have so . We have and where . We deduce by induction using Lemma 5.28 that .

6.3.Nested sequences

In this subsection, we assume that is admissible. For we say that a -admissible number is -nested if we have for all . We write for the class of -nested numbers. For we simply say that is -nested and we write .

Definition 6.10. We say that is nested if for all , we have

Note that the inclusion always holds. In [5, Section 8.4], we gave examples of nested and admissible non-nested sequences in the case of transseries, i.e. with for all . We next give an example in the hyperserial case.

Example 6.11. We claim that the sequence from Example 6.6 is nested. Indeed, let and . We have for a certain with . Let us check that the conditions of Definition 6.4 are satisfied for .

First let . We want to prove that . We have for a certain . Now , so .

Secondly, let . We want to prove that . We have for a certain . Then by the previous paragraph. Now so .

Finally, we claim that . This is immediate since the dominant term of is positive infinite, so . Therefore is nested.

A crucial feature of nested sequences is that they are sufficient to describe nested expansions. This is the content of Theorem 6.15 below.

Lemma 6.12. Let . If , or and is not tail-atomic, then the hyperserial expansion of is

If , is tail-atomic, and is a hyperserial expansion, then and the hyperserial expansion of is

Proof. Recall that . By Corollary 6.8, we have ,. So we may assume without loss of generality that .

We claim that . Assume for contradiction that and write accordingly. Then Corollary 5.6 implies that , in which case we define , or for some ordinal and for some . Therefore , so . This implies that

Recall that . Assume that , so . Since is -atomic, we also have . Let be minimal with or . We have and . In particular, the number is log-atomic. If , this contradicts the fact that . If , then implies

But then is not a monomial: a contradiction. Assume now that . So and . But then is not defined: a contradiction. We conclude that .

If , or if and is not tail-atomic, then our claim yields the result. Assume now that and that is tail-atomic where , and is a hyperserial expansion. Then the hyperserial expansion of is .

We next show that . If , then , and we conclude with Lemma 6.7 that . Assume for contradiction that . Since is log-atomic, we must have . By the definition of coding sequences, this implies that and . So , whence , , and . In particular the number is -atomic, hence tail-atomic. Since , the claim in the second paragraph of the proof, applied to , gives . But then also : a contradiction.

We pursue with two auxiliary results that will be used order to construct a infinite path required in the proof of Theorem 6.15 below.

Lemma 6.13. For , there is a finite path in with or .

Proof. By Lemma 5.16, it is enough to find such a path in . Write . Assume first that , so and . If is not tail-atomic, then the hyperserial expansion of is and is the dominant term of for some . Then the path with and satisfies . If is tail-atomic, then there exist , and such that the hyperserial expansion of is . Let be a term in with . Then the path with and satisfies .

Assume now that . In view of (3.6), we recall that there are an ordinal and a number with

If is a limit ordinal, then by Lemma 6.12, we have a hyperserial expansion . Let and set and , so that is a path in . By Lemma 5.15, there is a subpath in , hence also a path in , with . So . If is a successor ordinal, then we may choose for a certain . By Lemma 6.12, we have a hyperserial expansion . As in the previous case, there is a path in with , whence .

Corollary 6.14. For and , there is a finite path in with and or .

Proof. This is immediate if . Assume that the result holds at and pick a corresponding path with (resp. ). Note that the dominant term of (resp. ) lies in by Lemma 6.7. Moreover is a term of (resp. ). By the previous lemma, there is a path in with or , so satisfies the conditions.

Theorem 6.15. There is a such that is nested.

Proof. Assume for contradiction that this is not the case. This means that the set of indices such that we do not have is infinite. We write where . Fix and let . Let such that

(6.2)

let and let be any finite path with

We claim that we can extend to a path with , and such that is a bad index in . Indeed, in view of Definition 6.4 for , the relation (6.2) translates into the following three possibilities:

Consider a and the path in . So is a finite path with . Thus there exists a path which extends with , where is a bad index in . Repeating this process iteratively for , we construct a path that extends and such that and such that is a bad index in . At the limit, this yields an infinite path in that extends each of the paths . This path has a cofinal set of bad indices, which contradicts Theorem 1.1. We conclude that there is a such that is nested.

Lemma 6.16. Assume that is nested. Then we have .

Proof. Note that . The result thus follows from Corollary 6.8 and the assumption that is nested.

Lemma 6.17. Assume that is nested. Let , and with

(6.3)

for a certain with and whenever . If , then we have

Proof. The proof is similar to the proof of Lemma 5.31. We have and we must have since . If follows from the deconstruction lemmas in Section 5.3 that . This proves the result in the case when .

Now assume that . Setting , let us prove by induction on that

For , the last relation yields the desired result.

If , then we have by assumption and we have shown above that . We have and is a monomial, so (6.3) yields . This deals with the case . In addition, we have because and . Let us show that

(6.4)

If , then this follows from the facts that and . If and , then . If , then and , so .

Assume now that and that the induction hypothesis holds for all smaller . We have

(6.5)

Since is nested, we immediately obtain , whence as above. Since and is nested, we have . Using (6.5), (6.4), and the decomposition lemmas, we observe that the relation is equivalent to

(6.6)

We have , so . Note that

So it is enough, in order to derive (6.6), to prove that . Now

by Lemma 6.9, whence by Lemma 5.25.

For , and , we have by Lemma 6.16. We may thus consider the strictly increasing bijection

We will prove Theorem 1.2 by proving that the function group on generates the class , i.e. that we have . We first need the following inequality:

Lemma 6.18. Assume that is nested. Let with and let . On , we have if and if .

Proof. It is enough to prove the result for . Assume that . Let and set , so that

Note that

If , then and is strictly increasing. So we only need to prove that , which reduces to proving that . Let be the dominant term of . Our assumption that is nested gives , whence . We deduce that . Lemma 5.27 implies that is -truncated.

and implies that

is a strictly positive term. We deduce that , whence . The other cases when or when are proved similarly, using symmetric arguments.

We are now in a position to prove the following refinement of Theorem 1.2.

Theorem 6.19. If is nested, then is a surreal substructure with .

Proof. By Proposition 4.1, the class is a surreal substructure, so it is enough to prove the equality. We first prove that .

Assume for contradiction that there are an and a , which we choose minimal, such that cannot be written as where is a hyperserial expansion. Set , and .

Our goal is to prove that there is a number and with

(6.7)

Assume that this is proved and set . The first condition and Lemma 6.16 yield and the relations and . The second and third condition, together with Lemma 6.17, imply . The first condition also implies that : a contradiction. Proving the existence of and is therefore sufficient.

If or and , then and satisfy (6.7). Assume now that and that , whence . If then and satisfy (6.7). Assume therefore that . This implies that there exist and with By the definition of coding sequences, there is a least index with or , so

We have and . So by Corollary 5.6, we must have for a certain and for a certain . Note that . Recall that and , so . The case cannot occur for otherwise

would not lie in . So . Let and

We have and , so and satisfy (6.7). We deduce that is a subclass of .

Conversely, consider and set . So there are and with . Let . By Lemma 6.18, there exist with and , whence . Since is strictly monotonous, we get . The numbers and are monomials, so . Therefore .

In view of Theorem 6.19, Lemma 6.18, and Proposition 4.1, we have the following parametrization of :

We conclude this section with a few remarkable identities for .

Lemma 6.20. If is nested, then for and , we have .

Proof. By [5, Lemma 4.5] and since the function is strictly monotonous, it is enough to prove that . By induction, we may also restrict to the case when . So assume that . Recall that by Lemma 6.9. Since , we deduce with Lemma 5.25 that . It follows using the decomposition lemmas that .

Proposition 6.21. If is nested, then we have .

Proof. We have by definition of . So we only need to prove that . Consider . Since is nested, the number is -admissible, so we need only justify that . Since is -admissible, we have . But is -nested, so for a certain term . We deduce that , whence .

Assume for contradiction that and write where and . Note that : otherwise and would be zero for all , thereby contradicting Definition 6.1(e). By Corollary 5.6, we must have for a certain ordinal and for a certain . Consequently, . If , then the condition implies , which leads to the contradiction that . If , then , whence : a contradiction.

Corollary 6.22. If is nested, then for , we have

Corollary 6.23. If is nested and , then

Proposition 6.24. Assume that is nested with , assume that and write . Consider the coding sequence with for all , with the only exception that

If , or and , then is nested and we have

where is the class of -nested numbers.

Proof. Assume that , or and . In particular, if is -admissible, then , so . For , it follows that is -admissible if and only if is -admissible. Let be the class of -admissible numbers, for each . We have by the previous remarks, and is admissible. For , we have , so

Moreover, , so

So is nested. We deduce that , that is, we have a strictly increasing bijection . It is enough to prove that for with , we have . Proceeding by induction on , we may assume without loss of generality that . By [6, identity (6.3)], the function has the following equation on :

So it is enough to prove that . Note that and where . So , whence . This concludes the proof.

6.4.Pre-nested and nested numbers

Let be a number. We say that is pre-nested if there exists an infinite path in without any bad index for . In that case, Lemma 6.2 yields a coding sequence which is admissible due to the fact that with the notations from Section 6. By Theorem 6.15, we get a smallest such that is nested. If , then we say that is nested. In that case, Theorem 6.19 ensures that the class of -nested numbers forms a surreal substructure, so can uniquely be written as for some surreal parameter .

One may wonder whether it could happen that . In other words: do there exist pre-nested numbers that are not nested? For this, let us now describe an example of an admissible sequence such that the class of -nested numbers contains a smallest element . This number is pre-nested, but cannot be nested by Theorem 6.19. Note that our example is “transserial” in the sense that it does not involve any hyperexponentials.

Example 6.25. Let be a nested sequence with . Let be the simplest -nested number. We define a coding sequence by

Note that

where is an infinite monomial, so is -nested. In particular, the sequence is admissible.

Assume for contradiction that there is a -nested number with . Since , we have . Recall that and are purely large, so . In particular

which contradicts the assumption that is -nested. We deduce that is the minimum of the class of -nested numbers. In view of Theorem 6.19, the sequence cannot be nested.

The above examples shows that there exist admissible sequences that are not nested. Let us now construct an admissible sequence such that the class of -nested numbers is actually empty.

Example 6.26. We use the same notations as in Example 6.25. Define and set for all . We claim that the coding sequence is admissible. In order to see this, let . Then

So is -admissible [why exactly?], whence is admissible. Assume for contradiction that is non-empty, and let . Then is -nested, so , whence : a contradiction.

7.Numbers as hyperseries

Traditional transseries in can be regarded as infinite expressions that involve , real constants, infinite summation, exponentiation and logarithms. It is convenient to regard such expressions as infinite labeled trees. In this section, we show that surreal numbers can be represented similarly as infinite expressions in that also involve hyperexponentials and hyperlogarithms. One technical difficulty is that the most straightforward way to do this leads to ambiguities in the case of nested numbers. These ambiguities can be resolved by associating a surreal number to every infinite path in the tree. In view of the results from Section 6, this will enable us to regard any surreal number as a unique hyperseries in .

Remark 7.1. In the case of ordinary transseries, our notion of tree expansions below is slightly different from the notion of tree representations that was used in [30, 38]. Nevertheless, both notions coincide modulo straightforward rewritings.

7.1.Introductory example

Let us consider the monomial from Example 5.9. We may recursively expand as

In order to formalize the general recursive expansion process, it is more convenient to work with the unsimplified version of this expression

Introducing as a notation for the “power” operator, the above expression may naturally be rewritten as a tree:

In the next subsection, we will describe a general procedure to expand surreal monomials and numbers as trees.

7.2.Tree expansions

In what follows, a tree is a set of nodes together with a function that associates to each node an arity and a sequence of children; we write for the set of children of . Moreover, we assume that contains a special element , called the root of , such that for any there exist a unique (called the height of and also denoted by ) and unique nodes with ,