Surreal substructures

Vincent Bagayoko

CNRS, LIX

Campus de l'École polytechnique

1, rue Honoré d'Estienne d'Orves

Bâtiment Alan Turing, CS35003

91120 Palaiseau, France

Département de Mathématique

Université de Mons, Le Pentagone

20, Place du Parc

B-7000 Mons, Belgique

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

October 30, 2022

Abstract

Conway's field of surreal numbers comes both with a natural total order and an additional “simplicity relation” which is also a partial order. Considering as a doubly ordered structure for these two orderings, an isomorphic copy of into itself is called a surreal substructure. It turns out that many natural subclasses of are actually of this type. In this paper, we study various constructions that give rise to surreal substructures and analyze important examples in greater detail.

1Introduction

1.1Surreal numbers

The class of surreal numbers was discovered by Conway and studied in his well-known monograph On Numbers and Games [11]. Conway's original definition is somewhat informal and goes at follows:

“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 magic of surreal numbers lies in the fact that many traditional operations on integers and real numbers can be defined in a very simple way on surreal numbers. Yet, the class turns out to admit a surprisingly rich algebraic structure under these operations. For instance, the sum of two surreal numbers and is defined recursively by

(1.1)

In section 3 below, we recall similar definitions for subtraction and multiplication. Despite the fact that the basic arithmetic operations can be defined in such an “effortless” way, Conway showed that actually forms a real-closed field that contains . Strictly speaking, some care is required here, since the surreal numbers form a proper class. In particular, it contains all ordinal numbers . We refer to appendix B for ways to deal with this kind of set-theoretic issues.

One convenient way to rigourously introduce surreal numbers is to regard them as “sign sequences” indexed by the elements of an ordinal number , called the length of : see section 2.1 below for details. Every ordinal itself is represented as with for all . The number is represented by the sign sequence of length . The ordering on corresponds to the lexicographical ordering on sign sequences, modulo zero padding when comparing two surreal numbers of different lengths. The sign sequence representation also induces the important notion of simplicity: given , we say that is simpler as , and write , if the sign sequence of is a truncation of the sign sequence of . The simplicity relation is denoted by in some previous works [8, 27, 3].

The sign sequence representation was introduced and studied systematically in Gonshor's book [21]. As we will see in section 3, it also allows for a natural extension of ordinal arithmetic to the surreal numbers. In order to avoid confusion, we will systematically use the notations and for ordinal sums and products and for ordinal exponentiation. For instance, in , we have . Given an ordinal , it is also natural to define the set of all surreal numbers of length . It turns out that is a real-closed subfield of if and only if is an -number, i.e. [12, Proposition 4.7 and Corollary 4.9].

1.2Exponentiation, derivation, and hyperseries

Quite some work has been dedicated to the extension of basic calculus to the surreal numbers and to the study of various operations in terms of sign sequences. In his book [21], Gonshor shows how to extend the real exponential function to . This exponential function actually admits the same first order properties as the usual exponential function: the class is elementarily equivalent to as an exponential field. In fact, they are even elementarily equivalent as real exponential ordered fields equipped with restricted analytic functions [12, Theorem 2.1]. Here we recall that a restricted real analytic function is a power series at the origin that converges on a small closed ball with . Then it can be shown that the definition of extends to surreal numbers with .

Another important question concerns the possibility to define a natural derivation on the surreal numbers, which is non-trivial in the sense that . Such a derivation was first constructed by Berarducci and Mantova [8], while making use of earlier work by van der Hoeven and his student Schmeling [35]. It was shown in [3] that this “Italian” derivation has “similarly good properties” as the exponential function in the sense that is elementary equivalent to the field of transseries as an H-field. Here transseries are a generalization of formal power series. They form an ordered exponential field that comes with a derivation. The notion of an H-field captures the algebraic properties of this field as well as those of so-called Hardy fields. We refer to [1] for more details.

The above results on the exponential function and the Italian derivation on rely on yet another representation of surreal numbers as generalized power series with real coefficients and monomials such that is simpler than any other with the same valuation as : see section 2.3 for details. Indeed, ordinary power series and Laurent series in can be regarded as functions in , so they come with a natural derivation. More generally, the exponential function on makes it possible to interpret any transseries in as a surreal number, which makes it again possible to derive such surreal numbers in a natural way.

Unfortunately, not all surreal numbers are transseries in . For instance, the surreal number is larger than any transseries in . In order to be able to intepret all surreal numbers as functions in , two ingredients are missing: on the one hand, we need to introduce ordinal “iterators” of the exponential function that grow faster than finite iterates. For instance, we have . On the other hand, we need to be able to represent so-called nested transseries such as

(1.2)

The present paper is part of an ongoing project to represent any surreal number as a generalized “hyperseries” in , which takes these observations into account. This project was first mentioned in [26] and further detailed in [2]. For progress on the “series side”, we refer to [23, 35, 26, 13]. The derivation cannot be compatible with a composition law on [9, Theorem 8.4]. More specifically, it was noted in [2] that the Italian derivation fails to satisfy for all . Ultimately, the ability to represent surreal numbers as hyperseries evaluated at should lead to compatible definitions of a derivation and a composition on .

1.3Surreal substructures

In the course of the above project to construct an isomorphism between and a suitable class of hyperseries, one frequently encounters subclasses of that are naturally parameterized by itself. For instance, Conway's generalized ordinal exponentiation is bijective, which leads to a natural parameterization of the class of monomials by (see Theorems 5.2 and 5.11). Similarly, nested expressions such as (1.2) do not give rise to a single surreal number, but rather to a class of surreal numbers that is naturally parameterized by (see Theorem 8.8). Yet another example is the class of log-atomic surreal numbers that occurs crucially in the construction of derivations on [8, Section 5.2].

In these three examples, the parameterizations turn out to be more than mere bijective maps: they actually preserve both the ordering and the simplicity relation . This leads to the definition of a surreal substructure of as being an isomorphic copy of inside itself. Surreal substructures such as , , and behave similarly as the surreal numbers themselves in many regards. In our project, we have started to exploit this property for the definition and study of new functions on such as hyperlogarithms and nested transseries.

The main goal of the present paper is to develop the basic theory of surreal substructures for its own sake and as a new tool to study surreal numbers. We hope to convey the sense that surreal substructures are at the same time very general and very rigid subclasses of and that several problems regarding the enriched structure of (highlighted in particular in the work of Gonshor [21], Lemire [28, 29, 30], Ehrlich [16, 15, 17], Kuhlmann–Matusinski [27], Berarducci–Mantova [8], and Aschenbrenner–van den Dries–van der Hoeven [3]) crucially involve surreal substructures. Even for very basic subclasses of , such as , we suggest that it deserves our attention when they form surreal substructures.

A substantial part of our paper (namely, sections 4, 5, and 6) is therefore devoted to basic but fundamental results. Some of these general facts were known and rediscovered in different contexts [31, 16]. However, they mainly appeared as auxiliary tools in these works. In this paper, we aim at covering the most noteworthy facts in a self-contained and organized way. In the course of our exposition, we identify which properties of surreal substructures are systematic and which ones are proper to specific structures. We also include a wide range of examples. This effort culminates in the last two sections 7 and 8, where we present the examples that motivated our paper and that are important for our program to construct an isomorphism between and the class of hyperseries. We refer to [5] for some first applications in this direction. In Appendix A, we also compiled a small atlas for the most prominent examples of surreal substructures.

1.4Summary of our contributions

Let us briefly outline the structure of the paper. In section 2, we recall the three main representations of surreal numbers. In section 3, we recall the definitions of basic arithmetic operations on surreal numbers. We also show how to extend the ordinal sum and the ordinal product to .

In section 4, we introduce surreal substructures, our main object of study, as isomorphic copies of inside itself. Any surreal substructure comes with a defining isomorphism that is unique and that we consider as a parameterization of the elements in by . Defining isomorphisms and can be composed to form the defining isomorphism of a new surreal structure that we call the imbrication of inside . More generally, we will often switch between the study of surreal substructures and that of their parameterizations. A consequent part of section 4.1 is a reformulation of notions and arguments found in [31, 16, 17]; see Remark 4.8.

In section 5, we investigate the existence of fixed points for the defining isomorphism of a given surreal substructure . More precisely, we give conditions on under which the class of such fixed points is itself a surreal substructure. Determining the class allows us in some cases to compare the defining isomorphisms of two surreal substructures. This task leads us to study surreal substructures which are closed under non-empty, set-sized suprema in of chains in . Such a surreal substructure is said -closed, and has the following properties:

In fact any well-ordered sequence of -closed surreal substructures can be similarly “imbricated”, and thus -closed surreal substructures can be seen as words in a rich language that captures at the same time the notions of fixed points, imbrications and intersections of surreal substructures. One direct application is a new proof of a theorem by Lemire [29]; see Remark 5.17.

In section 6, we study subclasses whose elements are the simplest representatives of members in a convex partition of a surreal substructure . Under a set-theoretic condition on , we prove that this class forms a surreal substructure of (Theorem 6.7) whose parameterization admits a short recursive definition. A weaker version of this theorem was first proved by Lurie [31]; see Remark 6.8. A particularly important special case is when the convex partition is induced by a group action (see section 6.3). We also introduce the notion of a sharp convex partition of a surreal substructure which makes closed within (Theorem 6.14).

Our final sections 7 and 8 concern the application of our results to some prominent examples of specific surreal substructures. This includes the structure of purely infinite surreal numbers of [21], the structure of monomials of [11], the structure of log-atomic numbers of [8], the structure of -numbers of [27], and various structures of nested monomials, including . Our results about nested monomials in section 8 are analogous to Lemire's work on continued exponential expressions [30], when replacing ordinal exponentiation by traditional exponentiation. The appendix A contains a short overview of the surreal substructures encountered in this paper.

1.5Notations

We will systematically use a bold type face to denote classes such as that may not be sets. Given a partially ordered class and subclasses of , we write if for all and . This holds in particular whenever or . For elements of , we write and instead of and . Given more than two subclasses of , we also write whenever for all .

If , we let denote the class of elements with . In the special case when is an ordered monoid, we simply write and .

We use similar notations for non-strict orders .

2Different presentations of surreal numbers

Surreal numbers can be represented in three main ways: as sign sequences, as generalized Dedekind cuts, and as generalized power series over . In this section, we briefly recall how this works, and review the specific advantages of each representation. We refer to [11, 21, 16, 15, 32] for more details.

2.1Surreal numbers as sign sequences

The sign sequence representation is most convenient for the rigourous development of the basic theory of surreal numbers. It was introduced by Gonshor [21, page 3] and we will actually use it to formally define surreal numbers as follows:

Definition 2.1. A surreal number is a map , where is an ordinal number. We call the length of and the map the sign sequence of . We write for the class of surreal numbers.

It follows from this definition that is a proper class. Given a surreal number , it is convenient to extend its sign sequence with zeros to a map and still denote this extension by . In other words, we take for all . Given and , we also introduce its restriction to as being the zero padded restriction of the map to : we set for and for .

The first main relation on is its ordering . We define it to be the restriction of the lexicographical ordering on the set of all maps from to . More precisely, given distinct elements , there exists a smallest ordinal with . Then we define if and only if .

The second main relation on is the simplicity relation : given numbers , we say that is simpler than , and write , if . We write for the set of surreal numbers that are strictly simpler than . The partially ordered class is well-founded, and is well-ordered with order type .

Every linearly ordered—and thus well-ordered—subset of has a supremum in : for any and , one has ; for any with all , one has . We will only consider suprema in and never in . Numbers that are equal to are called limit numbers; other numbers are called successor numbers. Limit numbers are exactly the numbers whose length is a limit ordinal.

2.2Surreal numbers as simplest elements in cuts

If are sets of surreal numbers satisfying , then there is a simplest surreal number, written , which satisfies [21, Theorem 2.1]. We call the Conway bracket. Notice that is the simplest such number in the strong sense that for all with , we have . The converse implication may fail: see Remark 4.21 below.

Now consider two more sets of surreal numbers with . If has no strict upper bound in and has no strict lower bound in , then we say that is cofinal with respect to . We say that and are mutually cofinal if they are cofinal with respect to one another, in which case it follows that . These definitions naturally extend to pairs of classes with . Note however that is not necessarily defined for such classes. Indeed, there may be no number with (e.g. for and ).

We call a pair of sets with a cut representation of . Such representations are not unique; in particular, we may replace by any mutually cofinal pair . For every surreal number , we denote

which are sets of surreal numbers. We call and the sets of left and right options for . By [21, Theorem 2.8], one has and the pair is called the canonical representation of .

This identity is the fundamental intuition behind Conway's definition of surreal numbers precisely as the simplest numbers lying in the “cut” defined by sets of simpler and previously defined surreal numbers. Of course, this is a highly recursive representation that implicitly relies on transfinite induction.

Conway's cut representation is attractive because it allows for the recursive definition of functions using by well-founded induction on or its powers. For instance, there is a unique bivariate function such that for all , , we have

(2.1)

Here we understand that denotes the set and similarly for . This recursive definition is justified by the fact that the elements of the sets , , and are all strictly simpler than for the product order on . This precise equation is actually the one that Conway used to define the addition on . We will recall similar definitions of a few other arithmetic operations in section 3 below.

2.3Surreal numbers as well-based series

Let be a field and let be a totally ordered multiplicative group for the ordering . A subset is said to be well-based if it is well-ordered for the opposite ordering of (i.e. there are no infinite chains in ). A well-based series in and over is a map whose support is a well-based subset of . Such a series is usually written as , where and the set of all such series is denoted by . Elements in and are respectively called coefficients and monomials. We call the monomial group. The support of any non-zero element admits a largest element for , which is called the dominant monomial of and denoted by .

It was shown by Hahn [22] that forms a field for the natural sum and the usual Cauchy convolution product

In , there is also a natural notion of infinite sums: if is a set and is a family of well-based series in , then we say that it is summable if is well-based and is finite for every . In that case, we define the sum of this family by

Consider a second monomial group and a map . We say that is strongly linear if it is -linear and for every summable family in , the family is summable in with . By [25, Proposition 10], in order to show that a linear map is strongly linear, it suffices to prove that the above condition holds for families of scalar multiples of monomials. So is strongly linear if and only if for all , the family is summable, with

Since the support of any is well-based, the order type of for the opposite order of is an ordinal. Now consider an -number . We recall that this means that , where stands for Cantor's -th ordinal power of . It is known [20, Corollary 6.4] that the series with form a subfield of .

The ordering on induces a natural valuation on whose residue field is . The Archimedean class of a non-zero surreal number is the class of all with the same valuation as . One of the discoveries of Conway was that admits a simplest element that we will denote by . Let be the class of all that we may obtain in this way. Conway also constructed an order preserving bijection that extends Cantor's ordinal exponentiation.

Through this -map and the so-called Conway normal form [11, Chapter 5], it turns out that the field is naturally isomorphic to a field of well-based series , for which becomes the monomial group. For this series representation, any number has a set-sized support . The Conway normal form of coincides with its expression as a series . For we sometimes write instead of in order to indicate that we have , and thus that is a truncation of as a series.

3Arithmetic on surreal numbers

In the sequel of this paper, by “number”, we will always mean “surreal number”.

3.1Surreal arithmetic

We already explained the usefulness of Conway's cut representation for the recursive definition of functions on and mentioned the addition (2.1) as an example. In fact, one may define all basic ring operations in a similar way:

(3.1)
(3.2)
(3.3)
(3.4)
(3.5)

One major discovery of Conway was that the surreal numbers actually form a real closed field for these operations and the ordering . As an ordered field, it naturally contains the dyadic numbers, which are the numbers with finite length, and the real numbers, which are the numbers of length whose sign sequence does not end with infinitely many consecutive identical signs.

The class of ordinals is also naturally embedded into by identifying an ordinal with the constant sequence of length with for all . Thus, in , expressions such as

make sense and are amenable to various computations and comparisons. See [11, Chapter 1] for more details on the field operations on . See [21, Chapters 1, 2 and 3] for more details on those operation in the framework of sign sequences and on the correspondence between cuts and sign sequences.

Using hints from Kruskal, Gonshor also defined an exponential function on , which we denote by [21, Page 145]. This function extends the usual exponential function on . In fact, it turns out that is an elementary extension of as an ordered exponential field [12, Corollary 5.5]. In other words, the usual exponential function and its extended version to satisfy the same first order properties over .

In order to define for using a recursive equation, one needs to find an appropriate characterization of the cut formed by inside the field generated by , , and . In exponential fields, the natural inequalities satisfied by such cuts involve truncated Taylor series expansions. Given and , let

If and is such that is already defined, then for , we should have

and one expects that such inequalities give sharp approximations of . Following this line of thought, Gonshor defined

(3.6)

The reciprocal of , defined on , is denoted . This also leads to a natural powering operation: given and , we define . Given , we have , but for more general elements , one does not necessarily have . (see [6] for more details).

3.2Extending ordinal arithmetic

We write and for the classes of non-zero and limit ordinal numbers, respectively. The class of ordinal numbers is equipped with two distinct sets of operations: Cantor's (non-commutative) ordinal arithmetic and Hessenberg's (commutative) arithmetic. For ordinals , we will denote their ordinal sum, product, and exponentiation by , and . Their Hessenberg sum and product coincide with their sum and product when seen as surreal numbers [21, Theorems 4.5 and 4.6]; accordingly, we denote them by and . We assume that the reader is familiar with elementary computations in ordinal arithmetic. In this section, we define operations on surreal numbers which extend ordinal arithmetic.

For numbers , we let denote the number, called the concatenation sum of and , whose sign sequence is the concatenation of that of at the end of that of . So is the number of length , which satisfies

(α<ℓ(x))
(β<ℓ(y))

It is easy to check that this extends the definition of ordinal sums. Moreover, the concatenation sum is associative and satisfies whenever and is a limit number.

We let denote the number of length , called the concatenation product of and , whose sign sequence is defined by

(α<ℓ(y),β<ℓ(x)).

Here we consider as a product in . Informally speaking, given and , the number is the -fold right-concatenation of with itself, whereas is the number obtained from by replacing each sign times by itself. We note that extends Cantor's ordinal product.

The operations and will be useful in what follows for the construction of simple yet interesting examples of surreal substructures. The remainder of this section is devoted to the collection of basic properties of these operations. The proofs can be skipped at a first reading, but we included them here for completeness and because we could not find them in the literature. We refer to [11, First Part] for a different extension of the ordinal product to the class of games (which properly contains ).

Lemma 3.1. For , we have

  1. .

  2. and .

  3. .

  4. if is limit.

Proof. ) Both and have length . Let and . Write where and . Then

) The numbers and have length . For , we have and .

) The number has length

Let and . If , then

Otherwise, there is such that and then

) The previous identities imply in particular that is linearly ordered by simplicity, which means that the supremum is well defined in . Assume is limit. If , then we have . Assume . Notice that we have , so

Let and . Since is a limit number, there is such that . Then

Remark 3.2. The previous lemma can be regarded as an alternative way to define the concatenation product. Yet another way is through the equation

(3.7)

Likewise, the contatenation sum has the following equation [15, Proposition 2]:

(3.8)

Note that these two equations are not uniform in the sense of Definition 4.29 below.

Proposition 3.3. Let .

  1. If , then if and only if .

  2. If , then if and only if .

Proof. a) If , then for with , Lemma 3.1(c) implies that

Conversely, if , then since , we may compute, for , the sign . We deduce that , so .

b) If , then given the maximal common initial segment of and , we have , with . Thus is strictly smaller than , which means that . Since the order is linear, this suffices to prove the result.

4Surreal substructures

4.1Surreal substructures and their parameterizations

Let be a subclass of and let be a family of ordering relations on . Then we say that a function is -increasing if is increasing for each with . If is also injective, then we say that it is strictly -increasing. If we have for all and , then we call an -embedding of into . We simply say that is an embedding if is a -embedding.

Definition 4.1. A surreal substructure is the image of an embedding of into itself.

Example 4.2. Given , the map is an embedding of into itself. If , then so is the map , by Proposition 3.3. Consequently:

Example 4.3. Let be an embedding of into itself with image . Then the map defines another embedding of into itself with image . In other words, if is a surreal substructure, then so is .

We claim that any strictly -increasing map is automatically an embedding. We first need a lemma.

Lemma 4.4. If are numbers such that and , then we have if and only if , and if and only if .

Proof. Since , we have if and only if there is with and . Now so and likewise holds if and only if there is with and . Notice that and imply that . In both cases, since , we have and . Therefore the existence of yields that of and vice versa. The other equivalence follows by symmetry.

Lemma 4.5. Assume that is a convex subclass of . Then every strictly -increasing function is an embedding .

Proof. Since is a linear order, the function is automatically an embedding for , so we need only prove that it is an embedding for . Assume for contradiction that there are elements of such that and . Let be the -maximal common initial segment of and . We have , so . Since is strictly -increasing, we have and , which given our assumption contradicts the previous lemma. Hence , which concludes the proof.

Since a surreal substructure is an isomorphic copy of into itself, it should induce a natural Conway bracket on . This actually leads to an equivalent definition of surreal substructures. Let us investigate this in more detail.

Let be an arbitrary subclass of . We say that is rooted if it admits a simplest element, called its root, and which we denote by . Given subclasses of , we let denote the class of elements such that . If is rooted, then we let denote its root. If and are sets, then we call the cut in defined by and . If for any subsets of the class is rooted, then we say that admits an induced Conway bracket.

Proposition 4.6. Let admit an induced Conway bracket. Then the map defined by

is an isomorphism .

Proof. We first justify that is well defined. Let be such that is well-defined and strictly -increasing on , with values in . We have where those sets are in so is a well-defined element of , and is strictly -increasing on . By induction, is a strictly increasing map . Let with , so that . By definition, the number is the simplest element with . Since and , it follows that . We deduce from Lemma 4.5 that is an embedding of into itself.

We now prove that by induction on for . Let be such that is a subset of . Let and where since is strictly -increasing and thus injective, the sets are uniquely determined and satisfy . Since admits an induced Conway bracket, the cut is rooted and contains , so . Since , we necessarily have . By induction, we conclude that .

Proposition 4.7. Let be a subclass of . Then is a surreal substructure if and only if it admits an induced Conway bracket.

Proof. Assume that admits an induced Conway bracket. By the previous proposition, is the range of the strictly -increasing function , whence is a surreal substructure. Conversely, consider an embedding of into itself with image . Let be subsets of and define . The function is strictly -increasing so , and we may consider the number . Now let . We have , so . Since is -increasing, this implies , which proves that , so admits an induced Conway bracket.

Remark 4.8. More generally, one may discard the existence condition for the Conway bracket and consider subclasses of that satisfy the following condition:

IN

For all subsets of with , the class is either empty or rooted.

A subclass satisfies IN if and only if there is a (unique) -initial subclass of and a (unique) isomorphism . This is in particular the case for the classes described in Section 6 below. For more details on this more general kind of subclasses, we refer to [16].

In this paper, we focus on surreal substructures. The characterizations given in Proposition 4.7 and Proposition 4.13 are known results. The second one was first proved (for more general types of ordinal sequences) by Lurie [31, Theorem 8.3], and both of them were proved by Ehrlich [16, Theorems 1 and 4].

Proposition 4.9. Let be a surreal substructure. The function is the unique surjective strictly -increasing function .

Proof. Let be a strictly -increasing function with image . By Lemma 4.5, it is an embedding. Given such that and coincide on , the numbers and of are both the simplest element of and are thus equal. It follows by induction that .

Lemma 4.10. Let be a surreal substructure. For , we have .

Proof. By Proposition 4.6, the map realizes an embedding of into , so the order type of the former is smaller than that of the latter, namely .

Given a surreal substructure , we call the defining surreal isomorphism of parametrization of . The above uniqueness property is fundamental; it allows us in particular to perform constructions on surreal substructures via their defining surreal isomorphisms and vice versa.

4.2Cut representations

Let be a surreal substructure. Given an element and subsets of with , we say that is a cut representation of in if . We refer to elements in and as left and right options of the representation. For , we write

and call this pair the canonical representation of in . We also write for the set .

A -final substructure of is a rooted final segment of for (and thereby necessarily a substructure). It is easy to see that this is the case if and only if is rooted and is the class of elements such that .

Proposition 4.11. Let be a surreal substructure and let and be cut representations in . For , we have

  1. if and only if and

  2. is a cut representation of in with respect to which any other cut representation of in is cofinal.

  3. .

Proof. The assertions ) and ) are true when by [21, Theorems 2.5 and 2.9]. By Proposition 4.6, the function is an isomorphism , satisfying the relation , so ) and ) hold in general. We have , since . Conversely, for and , we have and , so and have the same sign. We conclude that , which completes the proof of ).

4.3Imbrications

Let , be two surreal substructures. Then there is a unique -isomorphism that we call the surreal isomorphism between and . The composition is also an embedding, so its image is again a surreal substructure that we call the imbrication of into . We say that is a left factor (resp. right factor) of if there is a surreal substructure such that (resp. ).

By the associativity of the composition of functions, the imbrication of surreal substructures is associative. Right factors are determined by the two other substructures. More precisely, since is injective, the relation yields . The same does not hold for left factors:

Proposition 4.12. If are surreal substructures, then is a left factor of if and only if .

Proof. If , then . Assume that and let . We have where and , are respectively embeddings and so is an embedding . Hence is a surreal substructure with , which means that .

4.4Surreal substructures as trees

Through the identification , the class of surreal numbers can naturally be represented by a full binary tree of uniform depth , as illustrated in Figure 4.1 .

Figure 4.1. The class of positive surreal numbers as a tree. For clarity, only a few numbers up to the length are represented. Negative numbers are obtained through symmetry w.r.t. the -axis.

For each ordinal , we let denote the subtree of of nodes of depth , that is, the set of numbers with . This can be represented as the subtree obtained by cropping the picture at depth . In order to characterize surreal substructures in tree-theoretic terms, we need to investigate chains for : given a subclass , a -chain in is a linearly ordered (and thus well-ordered) subset of . If a -chain in admits a supremum in , we denote it . Note that the empty set has a supremum in if and only if has a root, in which case . We say that is the left successor of if and for every in . Right successors are defined similarly.

Proposition 4.13. Let be a class of surreal numbers. Then the following assertions are equivalent:

  1. is a surreal substructure.

  2. Every element of has a left and a right successor in and every -chain in has a supremum in .

Proof. Let be a surreal substructure. In , any element clearly admits a left successor and a right successor , and every -chain clearly admits a supremum. Since these properties are preserved by the isomorphism , we deduce ).

Assume now that ) holds. We derive ) by inductively defining an isomorphism . Applying ) to the empty chain, we note that the supremum of in is the minimum of for . So is rooted and we may define . Let be an ordinal such that is defined and strictly -increasing on . We distinguish two cases:

In both cases, this defines on and the extension is clearly strictly -increasing and strictly -increasing on every set for .

It remains to be shown that is strictly -increasing on . Given in , let be their -maximal common initial segment. We either have and thus , or and thus . So is strictly -increasing on .

By induction, the function is defined and -increasing on . Note that is well-founded since is well-founded and . By induction over , let us show that lies in the range of . If is the left or right successor of an element , then the induction hypothesis implies the existence of some with , and we get . Otherwise, we have where . We conclude that is an isomorphism.

Example 4.14. Consider the class defined by , , for all and and for every non-empty -chain without maximum in . It is easy to check that we have for every surreal number .

Example 4.15. Let . Then is isomorphic to , but is not a surreal structure. In other words, the condition ) cannot be replaced by the weaker condition that and be isomorphic.

The characterization ) gives us some freedom in constructing a surreal substructure: one only has to provide a mechanism for chosing left and right successors of already constructed elements, as well as least upper bounds for already constructed branches (i.e. -chains). Intuitively speaking, this corresponds to a way to “draw” as a full binary tree inside the binary tree that represents : see Figure 4.2 .

Figure 4.2. The (sub)tree representation of the surreal substructure (purple) from Example 4.14 within No (grey). The labels have the form . For instance .

4.5Convex subclasses

If are subclasses of , recall that is convex in if

and is -convex in if

We simply say that is convex (resp. -convex) if it is convex (resp. -convex) in . We let denote the convex hull of in , that is, for every number , we have if and only if and there are elements of such that . The convex hull of in is the smallest convex subclass of containing .

Lemma 4.16. Assume that is a surreal substructure. Then every non-empty convex subclass of is rooted.

Proof. In view of Propositions 4.6 and 4.7, it suffices to prove the lemma for . Let be a non-empty convex subclass of . Assume for contradiction that are two simplest elements with . Let be the smallest ordinal such that . Since and , we must have and . Now consider the number whose sign sequence is . Then , whence , but also ; a contradiction.

Lemma 4.17. If is a non-empty final segment of , then is the smallest ordinal in .

Proof. Given , we have , so C contains an ordinal. Let denote the smallest ordinal in . Given another ordinal , we have by minimality of . Since C is a final segment of No, it follows that . For any , we deduce that lies in the cut , whence . This shows that .

Proposition 4.18. Let be a surreal substructure.

  1. A convex subclass of is a surreal substructure if and only if it has no cofinal or coinitial subset.

  2. For subsets of , the cut is a surreal substructure.

  3. If is a surreal substructure, then is a surreal substructure.

  4. If is a surreal substructure, is a cut in and is strictly monotonic and surjective, then is a surreal substructure.

  5. The intersection of any set-sized decreasing family of surreal substructures that are convex in is a surreal substructure.

Proof. a) Assume that has no cofinal or coinitial subset and let be subsets of .

In each of the above cases, we have shown that is a non-empty convex subclass of . By Lemma 4.16, it is rooted. By Proposition 4.7, it follows that is a surreal substructure. Conversely, if is a surreal substructure, then given a subset of , we have

so is neither cofinal nor coinitial in .

b) This is a direct consequence of the previous point: the cut is by definition a convex subclass of , and given a subset of we have

By Proposition 4.7, it follows that is a surreal substructure.

c) Since is a surreal substructure, it has no cofinal or coinitial subset. It follows that the same holds for , which is thus a surreal substructure.

) We have is is increasing and if is decreasing. In both cases, is a cut in , hence a surreal substructure by ).

) Let be a linearly ordered set and let be decreasing for . Its intersection is convex. Let be a subset of . For , we have whence where . and . Writing and , we have . Moreover, for , we have so by convexity. This proves that and consequently that is neither cofinal nor coinitial in . Therefore is a surreal substucture by ).

Example 4.19. Cuts where are subsets of include -final substructures of and non-empty open intervals of , which are therefore convex surreal substructures. Note that non-empty convex classes of which are open in the order topology may fail to be surreal substructures. One counterexample is the class of finite surreal numbers, since it admits the cofinal subset .

Example 4.20. Here are some further examples and counterexamples of convex surreal substructures that we will consider later on.

Remark 4.21. For subsets of , the cut may fail to be a -final substructure of . In fact, by Proposition 4.11(c), it is a -final substructure of if and only if the canonical representation of in is cofinal with respect to , in which case we have .

Any convex subclass of is a generalized cut in where is the class of strict lower bounds of in and is the class of its strict upper bounds. However, those classes may not always be replaced by sets. In fact, the class is a cut with subsets of if and only if such sets can be found that are mutually cofinal with . The existence thus amounts to since cofinality is invariant under mutual cofinality (see the end of Appendix B for notes about cofinal well-ordered subsets).

Example 4.22. Recall that . Let for each and consider the class . Then is a convex surreal substructure of . Indeed, the sequence with is strictly decreasing and coinitial in . This shows that does not admit a coinitial subset. As a non-empty final segment of , the class also admits no cofinal subset. Proposition 4.18 thus implies that is a surreal substructure. We have , so is not a cut in .

4.6Cut equations

We already noted that the Conway bracket allows for elegant recursive definitions of functions on . Let us now study such definitions in more detail and examine how they generalize to arbitrary surreal substructures.

Definition 4.23. Let be surreal substructures. Let be functions defined for cut representations in and such that are subsets of whenever is a cut representation in . We say that a function has cut equation if for all , we have

We say that the cut equation is extensive if it satisfies

Note. We will see in the proof of Proposition 4.27 below that extensive cut equations preserve simplicity.

Example 4.24. A simple example of a cut equation is (3.3): . Here we have and we can take and . Note that this cut equation is extensive.

Taking and , and , we obtain the function with for all and for all .

See Example 4.32 below for more examples.

Remark 4.25. Our notion of cut equation is not restrictive on the function, since any function has cut equation with and . Thus it should not be confused with the notions of recursive definition in [19] and genetic definition in [34].

Example 4.26. Given sets of functions , cut equations of the form with

are extensive. We will write in this case. Note that it is common to consider well-defined cut equations of the form

where itself belongs to and .

Proposition 4.27. Let be surreal substructures. Let be strictly -increasing with extensive cut equation . Then is a surreal substructure, and we have .

Proof. We claim that is -increasing. Indeed, let with . We have , so and . We deduce by extensivity of that and , and thus . This implies that . Thus is strictly -increasing. So the composition is strictly -increasing. The function is an embedding by Proposition 4.6, so embeds into . In particular, is a surreal substructure. By Proposition 4.9, we conclude that .

As an application, we get the following well-known result (see [8, Proposition 4.22]).

Proposition 4.28. Let be a number, and let denote the class of numbers with . Then and are surreal substructures with

Proof. We have . By Proposition 4.18(b), this is a surreal substructure. Recall that for , we have . If , then we have so we may write

Seen as a cut equation in , this is an extensive cut equation, so by Proposition 4.27, we see that is a surreal substructure and that realizes the isomorphism .

Definition 4.29. Let be a function with cut equation . We say that is uniform at if we have

whenever is a cut representation of in . We say that is uniform if it is uniform at every .

Example 4.30. Let . The following cut equation for the function obtained from (3.8)

is uniform. On the contrary, the following cut equation for is not uniform:

Indeed, we have and , but .

Example 4.31. Let . By (3.7), the function has the following cut equation

which is uniform. On the contrary, the cut equation for is not uniform:

Indeed, if we were to apply this cut equation to the cut presentation of , then we would have as a left option and as a right option, which cannot be.

Example 4.32. Most common definitions of unary functions have known simple cut equations, and many of them are uniform, in particular throughout the work of H. Gonshor in [21]. For instance, the classical cut equations (3.3) and (3.6) for the functions and are uniform, so for and for any cut representation of in , we have

Example 4.33. We will also need an extension of the notion of uniform cut equation to functions . Specifically, by [21, Theorem 3.2], the classical cut equation (3.4) for the sum of two numbers is uniform in the sense that, given cut representations and of in , we have

(4.1)

Similarily for the multiplication, we have

where , , and range in , , and respectively.

Uniform cut equations have the interesting property that they can be composed.

Lemma 4.34. Let be surreal substructures. Let and be functions with uniform cut equations

Then has the uniform cut equation where for every cut representation in , we have and .

Proof. Let , let be a cut representation of in . By uniformity of the cut equation of at , we have

By uniformity of the cut equation of at , we have

whence the result.

Recall that a class is cofinal (resp. coinitial) with respect to a class if every element of has an upper bound (resp. lower bound) in . If , then we simply say that is cofinal (resp. coinitial) in .

Lemma 4.35. When are surreal substructures, the cut equation is uniform and extensive.

Proof. Let us first prove uniformity in the case when . Let be sets of surreal numbers and let . Since is strictly increasing and ranges in , the number is well defined and , which yields . Moreover, the set is cofinal in whereas is coinitial in , so . Hence and , which shows that the cut equation is uniform.

Now consider the general case and let be subsets of . Setting and , we have by uniformity of the cut equation for . Furthermore,

by uniformity of the cut equation for . Hence , which proves that is uniform. This cut equation has the form where are sets of functions, so it is extensive.

The above proposition shows that surreal isomorphisms satisfy natural extensive cut equations. Inversily, Proposition 4.27 shows that extensive cut equations give rise to surreal isomorphisms. As an application, if we admit that the operation

is well defined, then we see that it defines a surreal isomorphism. This is the parametrization of the class of monomials, that is, Conway's -map. This cut equation is also uniform (see [21, corollary of Theorem 5.2]), and we can for instance compute, for every number , the number

Whenever they exist, this shows the usefulness of extensive cut equations. Unfortunately, many common surreal functions such as the exponential do not admit extensive cut equations. The next proposition describes a more general type of cut equation that is sometimes useful.

Proposition 4.36. Let be surreal substructures. Let be a function from to the class of subsets of such that for with , the set is cofinal with respect to . For , let denote the class of elements of such that and are mutually cofinal. Let be an extensive cut equation on . Let be strictly increasing with cut equation

Then induces an embedding for each element of .

Proof. Let . If and satisfies , then is cofinal with respect to and hence to , and is cofinal with respect to and hence to , so . Therefore is a non-empty convex subclass of . Note that for , we have

For numbers lying in with , we have , which implies that . Since is a non-empty convex subclass of and is increasing and bijective, the class is a non-empty convex subclass of on which is strictly -increasing. By Lemma 4.5, the function induces an embedding and thus induces an embedding .

Example 4.37. A typical example is the following cut equation of [8, Theorem 3.8(1)] for the exponential function on the class of infinite monomials:

Here we have and .

5Fixed points

After introducing the -map as a way to parameterize the class of monomials, Conway remarks that for any ordinal , the number coincides with Cantor's -th ordinal power of . He then goes on with the definition of generalized -numbers as surreal numbers such that . It turns out that the class of generalized -numbers can be parameterized as well and actually forms a surreal substructure: see Conway's informal discussion [11, p 34–35] and Gonshor's formal proof [21, Theorem 9.1 and Corollary 9.2]. Gonshor gives further conditions for the class of fixed points of a surreal function to be a surreal substructure [21, Theorem 9.4].

In this section, we consider the more general problem of deciding, given a surreal substructure , whether admits fixed points, and possibly a whole surreal substructure of fixed points. A related fixed point theorem was obtained by Lurie [31, Theorem 8.2] in a somewhat different context.

5.1Fixed points and iterations of defining isomorphisms

For operators where are subclasses of and , it will be convenient to write for the -fold composition of with itself. In particular, .

Definition 5.1. Let be a surreal substructure. We say that a number is -fixed if . We let denote the class of -fixed numbers. Notice that is a subclass of .

If are surreal substructures with , then for every number , we have if and only if , and if and only if . In particular, the parametrizations of and coincide exactly on .

Proposition 5.2. If is a surreal substructure, then .

Proof. Let . For , we have , so . Assume for contradiction that is a proper subclass of , and consider with minimal length. For , let with . For all , we have , so by our minimality assumption and Lemma 4.10, we have .

Recall that is not -fixed, so . By symmetry, we may assume without loss of generality that , which implies that for all . For , let be the -maximal element of with . This element is well-defined since is a surreal substructure and . The number is -maximal in with , whence , so .

Since and , we have and . We deduce that and that . In particular, we have so , so is not -fixed, and we have .

Since for each , Lemma 4.10 implies . The latter decreasing sequence of ordinals is necessarily stationary; let be such that for all . By Lemma 4.10, it follows that for all , whence . But , which contradicts the minimality of . This absurdity completes our proof.

Example 5.3. Here are some examples of structures of fixed points where is a surreal substructure:

In general, the class may not be a surreal substructure. For instance, the class defined in Example-4.14 satisfies , and consequently has no fixed point. This raises the question of finding a condition on that will ensure to be a surreal substructure. One obvious first idea is to investigate when decreasing intersections of surreal substructures are surreal substructures.

5.2Closed subclasses

We introduce a notion of closed subclasses of an ambient surreal substructure . In the case when is a surreal substructure, we characterize its closedness in terms of its defining surreal isomorphism.

Definition 5.4. Let be a surreal substructure. Let be a subclass of . We say that is -closed, if the supremum in of any non-empty -chain in lies in .

Example 5.5.

The term “closed” suggests the existence of a topology. Indeed, we have:

Proposition 5.6. Let be a surreal substructure. Arbitrary intersections and finite unions of -closed subclasses of are -closed.

Proof. It is clear that and are -closed. Let be the intersection of a (possibly proper class-sized) non-empty family of -closed subclasses of . Let be a non-empty -chain in . We have for all , whence and is -closed.

Let be -closed subclasses of and let be a non-empty -chain in . If admits a -maximum, then . Otherwise, let be such that is -cofinal in . Then , so is -closed.

Lemma 5.7. If is a surreal substructure and is a -final substructure of , then is -closed.

Proof. The class is -final in , thus suprema of non-empty -chains in lie in .

It will sometimes be useful to comprehend closure in terms of projections.

Proposition 5.8. Let be a surreal substructure. A rooted subclass of is -closed if and only if every element of has a -maximal initial segment lying in .

Proof. Assume that is -closed. Consider with . Then the set of initial segments of lying in is non-empty and closed under taking suprema in . Consequently, indeed admits a -maximal initial segment in . Inversely, assume that is well defined on and let be a non-empty -chain in . If has a -maximum, then . Otherwise, , so . This shows that is -closed.

Definition 5.9. If is rooted and -closed, then we define to be the function that sends each element of to the -maximal initial segment of that lies in . It is by definition surjective, -increasing, and satisfies the relation . We call it the topological projection .

Since is -increasing when it exists, its fibers are -convex in .

Lemma 5.10. Let be surreal substructures and let be rooted. If is -closed and is -closed, then is -closed, and we have on .

Proof. Let . Since , we have , whence . The class is -closed so has a maximal initial segment lying in . Now is an initial segment of lying in , whence . We may thus consider the maximal initial segment of that lies in . If is simpler than , then , since . Similarly, , since . This proves that is the maximal initial segment of lying in .

We will mostly consider closures of surreal substructures in other ones. In this situation, closure can be regarded as a property of the defining surreal isomorphism:

Lemma 5.11. If are surreal substructures, then is -closed if and only if for any non-empty -chain of , we have .

Proof. Assume that the relation holds. Let be a non-empty -chain in and consider the set . Since is an -embedding, the set is a non-empty -chain in , whence (see Proposition 4.13). Our assumption on gives , so , and is -closed. Conversely, assume is -closed. Let be a non-empty -chain. Since is -increasing, the set is a non-empty -chain in , so , whence , which is the desired equality.

Lemma 5.12. Let be surreal substructures.

  1. If , then is -closed if and only if sends -closed subclasses of onto -closed subclasses of .

  2. If and are -closed, then so is .

  3. If and are -closed, then so is .

Proof. ) Assume is -closed and is a closed subclass of . Let be a non-empty -chain in . The set is a non-empty -chain in so its supremum lies in , and , so is -closed. Conversely, if sends closed classes of surreal numbers onto -closed subclasses of , then in particular is -closed.

) This is a direct consequence of ).

) Assume that and are -closed. Let be a non-empty -chain in . Then , and since is injective, we get , so is -closed.

We now come to the main interest of the notion of closure.

Proposition 5.13. Let be a limit ordinal. Let be a surreal substructure and let be a decreasing sequence of -closed surreal substructures of . Then its intersection is an -closed surreal substructure.

Proof. We use the characterization of surreal substructures given in Proposition 4.13. By Proposition 5.6, the class is -closed. In particular, the class has suprema of non-empty -chains. We also have which lies in by the -closure of each structure for , so the empty -chain has a supremum as well.

Let us now treat the case of left and right successors. Given , let and be the left and right successors of in , for each ordinal . For , we have and , so by the definition of left successors. Similarly, we get . Thus the sets and are -chains whose suprema in satisfy . For with and , we have so , whence . This means that is the right successor of in . Likewise, is the left successor of in . We conclude that is a surreal subtructure.

Corollary 5.14. If the surreal substructure is -closed, then is an -closed surreal substructure.

Proof. This is a direct consequence of Lemma 5.12, Proposition 5.13 and Proposition 5.2.

Remark 5.15. Corollary 5.14 is similar to [31, Theorem 8.2]. Lurie's result is more general, but when applied to an -closed surreal substructure , it only concludes that is a “good tree”. Good trees need not be surreal substructures. For instance,

is a good tree, but not a surreal substructure, since has two right successors and two left successors.

5.3Transfinite right-imbrications of surreal substructures

The class of -closed surreal substructures being closed under decreasing intersections, we are now in a position to define a notion of transfinite right-imbrications of -closed surreal substructures.

Theorem 5.16. Let be an ordinal. Let be a sequence of -closed surreal substructures. We define a sequence of -closed surreal substructures by the following rules:

Then each class for is an -closed surreal substructure, and if , then we have

(5.1)

Proof. We first need to prove that the definition is warranted. We do this by transfinite induction, while proving at the same time that the sequence is decreasing, and that each term is an -closed surreal substructure. Let be such that these assumptions hold strictly below . If is a successor ordinal, then and are -closed surreal substructures, whence is well defined and -closed (by Lemma 5.12). The surreal substructure is a left factor of , which implies that . If is limit, the intersection that defines is an -closed surreal substructure by Proposition 5.13, and is clearly decreasing.

We prove the identity (5.1) by induction on . Let be an ordinal such that (5.1) holds for any sequence and with . Let be such that . If for some ordinal , then

If is limit, then we have

(The injectivity of allowed us to move it through intersections).

Example 5.17. In [11, p 34–35], Conway informally discussed continued exponential expressions of the form

He outlined an approach for proving that the class of numbers that can be expressed in this way is order isomorphic to . Conway's ideas were rigorously worked out by Lemire [28, 29, 30]. He first proved the following result in the case when : given , let be the class of numbers such that there exists a sequence with

for all . Then is order isomorphic to . Moreover, writing for the isomorphism, has fixed points of any order , and the class of such fixed points is also order isomorphic to . This result follows from Theorem 5.16 by taking for all and . Then for all .

A similar result was proved by Lemire for more general continued exponential expressions [29, Theorem 4]. This result is more involved and presents similarities with our results about nested expansions in section 8 below.

Proposition 5.18. Let be an -closed surreal substructure. For each ordinal , let

Each is an -closed surreal substructure, and for , we have:

(5.2)
(5.3)

Proof. Most of this is a direct consequence of Theorem 5.16; we only need to prove the identity (5.3). Let be such that this identity holds for . Let be ordinal numbers with . Corollary 5.14 justifies that the same construction can be applied to the structure . If for , then we have

where we used (5.2) as well as the inductive hypothesis. If is limit, then

Note that for , the structure is the -fold imbrication of into itself, and we have . For , we have , by Proposition 5.2 and the identity (5.3). Thus transfinite right-imbrications of with itself allow us to define higher order fixed points of as being elements of the with . As we have seen, imbrication is left-distributive on decreasing intersections that form a surreal substructure. It is not right-distributive in general. For instance if is a proper -closed surreal substructure of , then is a proper subclass of .

Example 5.19. We will see in section 7.2 that the class coincides with .

Example 5.20. The class of fixed points of the -map was studied before in [11, 21, 31]; numbers in are called generalized -numbers. It also comes up in the study of the exponential function and the length of sign sequences [12, 27]. The class corresponds to a higher order fixed points of the -map and we expect it to play a similar role as for the study of the -th hyperexponential function.

6Convex partitions

Throughout this section, stands for a surreal substructure.

6.1Convex partitions

Definition 6.1. Let be a partition of into convex subclasses. We say that is a convex partition of . For we let denote the member of containing and recall that this class is rooted (by Lemma 4.16). We say that is -simple if , and we let denote the class of -simple elements of . For we write:

Remark 6.2. Convex partitions are sometime called condensations [33, Definition 4.1].

We can obtain as through the discrete partition with for all . Let for all . The map is a surjective, increasing projection. We refer to it as the -simple projection.

For the remainder of this subsection, let be a convex partition of . A quasi-order (or preorder) is a binary relation that is reflexive and transitive. The following lemma states basic facts on partitions of a linear order into convex subclasses.

Lemma 6.3. The relation is a linear quasi-order and restricts to a linear order on . For , we have if and only if .

Proof. It is well known that the partition corresponds to the equivalence relation on . The transitivity and irreflexivity of follow from that of on subclasses of . That its restriction to is a linear order is a direct consequence of the definition of and the equivalence stated above, which we now prove. If has only one member, then the result is trivial. Else let with . We have so . Conversely, assume that . Then which since is a partition implies that . For , there may be no element of such that for this would imply whence by convexity of this class: a contradiction. We thus have , that is, . By definition of , the relation implies that , whereas implies that , so , so .

For any subclass of , we let denote the class .

Lemma 6.4. Let , be subclasses of . Then the following statements are equivalent:

  1. .

  2. .

  3. .

Proof. All inequalities are vacuously true if or . Assume that and are non-empty and let and . Assume for contradiction that , but . Then there exist and with . By convexity of , this yields , whence . This contradiction shows that . The inverse implication clearly holds. The equivalence holds for similar reasons.

Lemma 6.5. For , the three following statements are equivalent:

  1. is -simple.

  2. There is a cut representation of in such that .

  3. .

Proof. Since is a cut representation of in , the assertion ) implies ).

Conversely, if is a cut representation of in with , then we have by the previous lemma. By Proposition 4.11(b), the cut representation is cofinal with respect to , so . Hence , again by Lemma 6.4. This shows that ) implies ).

Assume now that is -simple and let us prove ). For , we have , so , whence . We do not have since , so Lemma 6.3 yields , and in particular . This proves that , and similar arguments yield .

Assume finally that ) holds and let us prove a). We have so . Now the class is neither strictly greater nor strictly lower than , so our assumption imposes . We conclude that is -simple.

An order on a set is said to be dense if for any with , there exists a with .

Proposition 6.6. Assume that is dense. Then is the unique convex partition of such that is the class of -simple elements of .

Proof. For , let denote the class of elements of such that no -simple element lies strictly between and . The definition of the family only depends on the class , and not specifically on . For , we have .

Conversely, let , and assume for contradiction that lies outside of , say . Then and, being dense, there exists a -simple element between and . But implies , which contradicts the assumption that there is no simple element between and . We conclude that , which entails in particular that the partition is uniquely determined by .

If is dense, then we call the defining partition of . Notice that this is in particular the case when is a surreal substructure. We next consider a set-theoretic condition under which is always a surreal substructure.

We say that is thin if each member of has a cofinal and coinitial subset. For instance, the convex partition of where

is thin. Indeed each class for admits the cofinal and coinitial subset . See Example 6.15 below for more (counter)examples of thin convex partitions. If is thin, then we may pick a distinguished family such that each for is a cofinal and coinitial subset of , with . We write for any subclass of .

Theorem 6.7. If is thin, then is a surreal substructure. If is a cut representation in , then we have

Proof. Let be subsets of . For and , we have by Lemma 6.3. Therefore holds as well, which means that is well defined. Given and , there exists an with , since is cofinal in . It follows that , whence . A similar reasoning shows that for any . By Lemma 6.5, it follows that is -simple. Let be -simple. Given and , the -simplicity of , , and implies that , and in particular that . We deduce that , so . By Proposition 4.7, we conclude that the class is a surreal substructure.

Remark 6.8. The above theorem can be regarded as a strengthening of [31, Theorem 8.4] in a different framework. Indeed, Lurie's result is restricted to the case when and requires the additional assumption that

This condition is equivalent to the condition that be sharp in our terminology (see below); it fails for the partition of such that

which is the defining convex partition of the set of log-atomic numbers. Indeed, we have , where , but

Still, is a surreal substructure and even an -closed one.

When is thin, the structure is in addition cofinal and coinitial in , since for , we have . By the previous proposition, we may say that is thin if its defining partition is thin. If is not thin, then may fail to be a surreal substructure, but one can prove that there exists a unique -initial subclass of and a unique isomorphism between and .

For instance, we can obtain the ring of omnific integers of [11, Chapter 5] as where for each number , we set . This is not a surreal substructure since the cut is empty. Nevertheless, is -initial in . Note that different partitions may yield the same class (for instance replacing and with and respectively and leaving the other classes unchanged), in contrast to the case of dense partitions from Proposition 6.6. The partition in Example 6.15 below is not thin and yet is a surreal substructure.

Proposition 6.9. Assume that is thin. Then we have the following uniform cut equation for and :

Proof. The cut equation follows from Theorem 6.7 and the relation

Now towards uniformity, consider a cut representation of a number . We have so the number is well defined. Since is cofinal with respect to and is strictly increasing, the number lies in the cut , so . Conversely, we have , so . Since , we have , whence . We conclude that .

Corollary 6.10. If is thin and is a final segment of , then preserves ordinals.

Proof. If is an ordinal, then is a non-empty final segment of and thus of , so by Lemma 4.17, its simplest element is an ordinal.

For convex partitions of , we write if we have for every , and say that is finer than . If , then .

Recall that a directed set is a partial order such that for all , there exists a with .

Proposition 6.11. Let be a surreal substructure. Let be a non-empty directed set. If is a -increasing family of thin convex partitions of , then the intersection is a surreal substructure with defining thin partition given by

Proof. Given , the class is a non-empty convex subclass of and . Let be such that and let . Since is directed, there exists a in such that , whence . In particular, and . Since this is true for any , it follows that , so defines a convex partition of .

For , we have if and only if holds for all , so Lemma 6.5 implies . Now for , the set is cofinal and coinitial in , so is thin. Theorem 6.7 therefore implies that the class is a surreal substructure.

Proposition 6.12. Assume that is a final segment of and that are thin convex partitions of . Then for , we have , and in particular .

Proof. We prove the first inequality by induction on . Assuming that the inequality holds strictly below , we have

For , we have where , so , whence in particular . By Proposition 4.11(a), we have , whence the result by induction.

The second inequality is a consequence of the first one in the case when is the discrete partition of , which is -minimal and for which . Since is a final segment of , Proposition 4.17 gives . Moreover, for all with , we have , which yields by induction.

6.2Sharp convex partitions

We have encountered two different types of projections for surreal substructures. Given an -closed rooted subclass of a surreal substructure , the topological projection sends every element to the -maximal initial segment of lying in . Given a convex partition of the surreal substructure , the -simple projection sends to the unique -simple element lying in . It is natural to ask whether both types of projections relate to each other.

Given a surreal substructure and an -closed rooted subclass with , the topological projection is defined everywhere on . For each , we define . It is easy to see that defines a partition of into non-empty rooted -convex subclasses, and that is the class of roots where ranges in . The members of are not necessarily -convex in . For instance, one can prove that the structure is a -closed surreal substructure, with , for which contains and but not .

Conversely, given a convex partition of , the class may not be -closed, and when it is, it may be that and disagree. In some interesting cases, the projections and do coincide, and has additional properties, as we shall see now.

Definition 6.13. Let be a surreal substructure. We say that a convex partition of is sharp, if the canonical representation in of every -simple element is cofinal with respect to .

Assume that is thin and sharp. Then each element admits the cut representation in . By Proposition 4.11(b), this cut respresentation is mutually cofinal with . In view of Remark 4.21, we thus see that the sharpness is equivalent to the fact that the cut coincides with the -final substructure of for every . This corresponds to the notion of simple representation of [8, Definition 2.2]. We say that is sharp in if its defining partition is sharp.

The main interest of sharpness lies in the following equivalences:

Theorem 6.14. Let be a convex partition of the surreal substructure such that is a surreal substructure. The following statements are equivalent:

  1. is sharp.

  2. is -closed and .

  3. is -increasing.

  4. is -closed and is -increasing.

Proof. Assume that is sharp. Let us prove ), ) and ). Note that is -simple, whence . We know that when it exists is -increasing, and that is -increasing, so we need only prove that is -closed and .

Let be such that . We claim that is simpler than no element of . By symmetry, we may assume without loss of generality that . Since and is sharp, the set is cofinal with respect to . Assume for contradiction that we have for some . Let be such that and . Then . By Lemma 6.3, we also have , whence . It follows that , whence : a contradiction.

Since , our claim implies that is the maximal initial segment of any element of lying in , i.e. that is defined on and coincides with on this class. Since the classes cover , we see that is defined on , and . By Proposition 5.8, the structure is -closed.

We next prove that ) is a consequence of ). Assume for contradiction that is -closed with and that is not sharp. We treat the case when there are such that but has a strict upper bound in . Then , so , and . In particular, , whence : a contradiction. The other case is similar.

Assume next that is -increasing. For and such that , we have , so is the -maximal -simple initial segment of . This means that is -closed with topological projection . So ) implies ).

Assume is -closed and is -increasing. It follows that each fiber of where is convex for . As we have seen in the introduction of this section, we can construe as where for , we have . By Proposition 6.6, we have , so ) implies ). This concludes the proof.

Example 6.15. Convex partitions of a surreal substructure may or may not be sharp:

We end this subsection with two further properties of sharpness.

Proposition 6.16. Let be a non-empty directed set. Let be a -increasing family of thin convex partitions of a surreal substructure . If every with is sharp, then the defining thin partition of (defined in Proposition 6.11) is sharp.

Proof. We know by Proposition 6.11 that is a thin convex partition of with . Let . For and , there is such that where and . Since is sharp, there exists an with , so is cofinal with respect to . Likewise is coinitial with respect to , so is sharp.

Proposition 6.17. Let be a surreal substructure of that is also a final segment. Given a thin and sharp convex partition of , we have .

Proof. We already know from Corollary 6.10 that . Let be such that is an ordinal. The set is both empty and coinitial with respect to , which implies that and thus that is an ordinal.

6.3Group actions

In this subsection, we study one particularly important way in which convex partitions of surreal substructures arise, namely as convex hulls of orbits under a group action.

Let be a fixed surreal substructure. We define to be the (class-sized) group of strictly increasing bijections , with functional composition as the group law. Consider any set-sized subgroup of . Then naturally acts on through function application; we call a function group acting on .

Definition 6.18. We define the halo of an element under the action of by

Proposition 6.19. The classes for form a thin convex partition of .

Proof. Let . For any , we have . Indeed, we have for certain . Given , we also have for certain , whence , so that . We also have , whence and for any . The class is convex by definition. For , we know that contains , so the for form a convex partition of . For , the set is cofinal and coinitial in , so this partition is thin.

We write for the partition from Proposition 6.19 and say that an element of is -simple if it is -simple. We let denote the class of -simple elements. Proposition 6.19 implies that every property from Lemmas 6.3, 6.5 and 6.4 applies to the class of -simple elements. We call the -simple projection and write , , and instead of , and .

Proposition 6.20. is a surreal substructure with the following uniform cut equation in :

Proof. This is a direct consequence of Proposition 6.19, Theorem 6.7 and Proposition 6.9, where we take to be the required cofinal and coinitial subset of for each .

Remark 6.21. If is a set of strictly increasing bijective functions , we define to be the subgroup of generated by , i.e. the smallest subgroup of that contains . We say that is pointwise cofinal with respect to and we write if

This relation is transitive and reflexive. If , then , so . If and , then we say that and are mutually pointwise cofinal and we write . In that case, we have .

Let us now specialize Proposition 6.11 to group-induced convex partitions.

Proposition 6.22. Let be a non-empty directed set. If is a -increasing family of function groups acting on , then the function group generated by satisfies

Proof. If is -simple, then for , we have so is -simple. Conversely, assume is -simple for all . Then let where for , we have . Since is directed and is -increasing, there exists an index with and an element such that for all we have for all , and thus . Since is -simple, we have . This yields , so is -simple. This proves that .

Proposition 6.23. Let be a non-empty set, and let be a family of function groups acting on such that each is sharp in . Then where .

Proof. We have for the same reasons as above. Let . Let us prove by induction on that for , we have . By Lemma 6.5, this will prove that . For , the assertion is immediate. Assume therefore that and decompose , where . For every , we have . Since is -simple, the sharpness of implies that there exists an such that . By our inductive hypothesis, we have , so . The inequality is proved similarly.

Remark 6.24. The notions of thin convex partitions and function group actions are almost equivalent in the following sense. Let be a thin convex partition of , none of whose members has an extremum, and which satisfies the additional condition that there is a regular ordinal with for all . Then it can be shown that there is a group acting without global fixed points on such that . The converse also holds: for any function group acting without global fixed points on , we have for all .

7Common group actions

7.1Overview of known group actions

We conclude our study of surreal substructures with a closer examination of the action of various common types of function groups. We intentionally introduce these function groups without assigning specific domains; this will allow us to let them act on various surreal substructures.

Translations

Given , we define the translation by to be the map

The group acts in particular on and . More generally, if is a set-sized subgroup of , then acts on and .

Halos for the action of on are called finite halos and -simple elements correspond to purely infinite numbers. The class of purely infinite numbers is sometimes denoted ; see [11, 21].

Homotheties

Given , we define the homothety by the factor to be the map

The group acts in particular on , and . More generally, if is a set-sized subgroup of , then acts on , and .

Halos for the action of on are called archimedean classes and -simple elements are called monomials. The class of monomials is parameterized by the -map and forms a multiplicative cross section that is isomorphic to the value group of as a valued field (the valuation being induced by the ordering). The relations , , correspond to the asymptotic relations , , and from [26, 1]. Given , the projection coincides with the dominant monomial , when considering as a generalized series in .

Powers

Given , we define the -th power map by

Here and are the exponential and logarithm functions from section 3.1. The group acts in particular on and . More generally, if is a set-sized subgroup of , then the group acts on and .

Halos for the action of on are sometimes called multiplicative classes and -simple elements fundamental monomials. The class of fundamental monomials is parameterized by the -map: see [27, Proposition 2.5].

Exponentials

Writing

for all , we define

Both and act in particular on .

Halos and for the actions of and on are sometimes called levels and logarithmic-exponential classes respectively. The -simple elements are called log-atomic numbers and the class of such numbers is parameterized by the -map: see [8, Section 5]. The class of -simple elements is denoted by and parameterized by the -map: see [27, Section 3].

We notice that each of the above function groups is linearly ordered by

With the exception of , all these groups are also abelian. These are both strong properties which need not be imposed for the material of Section 6.3 to apply.

7.2Actions by translations

Throughout this subsection, let be a fixed set-sized subgroup of and let . If , then so . If , then given , the set is cofinal with respect to , so , whence .

Proposition 7.1. If acts on , then is an isomorphism.

Proof. We already know that is a -isomorphism so we only need to prove that it preserves sums. Let be such that preserves sums of elements lexicographically strictly simpler than . Recall that the addition is uniform in the sense that

Applying this to the cut equations given by Proposition 6.9 for , we obtain

and by uniformity of the cut equation for , we get

Thus . By induction, this proves that preserves sums of surreals and consequently that is an additive subgroup of .

Let us now focus on . By induction on , it is easy to see that and for all . In particular, this gives a description of in terms of sign sequences.

Let us next describe the structures for in terms of Conway normal forms and of -simplicity for some group acting on . By [12, Corollary 3.1], if is an ordinal, then the set is a subgroup of , which acts by translations on . If , then the sets and are mutually cofinal and coinitial, and , since . We claim that this generalizes to every ordinal.

Proposition 7.2. For , we have .

Proof. We proceed by induction on . The result obviously holds for . We saw that it holds for in Example 5.3. Assume that is a successor ordinal. Then the function is additive by Proposition 7.1, so is mutually cofinal and coinitial with . Let be -simple. Then is -simple, so the inductive hypothesis yields for a certain number . Since , we deduce that . Now for , there is with . We cannot have both and , so the contrapositive of Lemma 4.4 yields . Thus is -simple, so . Conversely, for , we have for a certain . We have , so . Similar arguments as above yield , whence . This proves that .

If is a limit ordinal, then Proposition 6.11 yields

A consequence of Propositions 7.1 and 7.2 is that is additive for all . In fact, we even have the following:

Proposition 7.3. For , the function is strongly linear, with .

Proof. Let and . Let us first show that for all and . By Proposition 7.2, the function is additive, so this holds for any dyadic number . In particular we have . Let be a non-dyadic real number. Let be such that for all . It is well known that contains only dyadic numbers. By Proposition 7.2 and (3.5), we have

where

The cut equation (3.5) for the surreal product by is uniform [21, Theorem 3.5], so

where

where respectively range in . Let us prove that and are mutually cofinal. Analog relations hold for the other sets so this will yield . Since is additive, for and , we have

Now and by our inductive hypothesis. Moreover, we have , since is dyadic. It follows that

Since is non-zero, we have , so this set is mutually cofinal with the set . Therefore is -linear.

Let us next prove by induction that . Let be such that for all . Let be an arbitrary cut representation in such that (resp. ) has no maximum (resp. minimum), so that (resp ) has no minimum (resp. maximum). Then we note that the cut equation

simplifies as

Considering the cut representation of , we deduce that we have

We have seen that is -linear, so the induction hypothesis yields

We thus have:

In particular preserves monomials.

Let be a number considered as a series in . By our previous arguments, the number is well defined. For all , we will write and . Let us prove by induction on the order type of that ; this will conclude the proof. The additivity and -linearity of yield the result for . If is successor and infinite, then has a minimum and , so

Assume now that is an infinite limit. Since is strictly increasing and monomial preserving, [21, Lemma 5.3] yields

where ranges over . Notice that the left (resp. right) options in the above representation of have no maximum (resp. minimum), so

Our inductive hypothesis yields

This concludes the proof.

Proposition 7.4. For , we have

In particular is a non-unitary subring of , and

Proof. The strong linearity of and the relation give

That this forms a (non-unitary) subring follows from the fact that is closed under addition, whence is closed under multiplication.

7.3Actions by homotheties

In this subsection, is a set-sized subgroup of and the defining isomorphism of . We will distinguish between confined and ample subgroups. We say that is confined if it is a subgroup of and ample if not. If is ample, then given , the maximum satisfies , which implies that is cofinal with respect to . Thus on , so . If is confined, then , so . For , natural examples of ample multiplicative subgroups include for , whereas natural examples of confined multiplicative subgroups include .

Remark 7.5. If is confined, then is -simple but is not coinitial with respect to which contains elements strictly below 1. So is not sharp in . The standard monomial group is sharp both in and in by [8, Corollary 4.17], but this observation does not generalize to arbitrary ample multiplicative subgroups of . For instance, if , then is -simple and but the number lies strictly above .

Proposition 7.6. Assume that is ample and let act on . Then the parameterization of is an isomorphism .

Proof. We only need to prove that is a morphism . Consider monomials with cut representations and such that , , and likewise for . Then [8, Proposition 4.19] yields

Given , this applies in particular to the cut representation of (and likewise for ) since is ample. We thus have

Note that . Assume and . Since is ample, there exists a such that