
. vincent.bagayoko@lilo.org
Logatomic numbers are surreal numbers whose iterated logarithms are monomials, i.e. additively irreducible numbers. Presenting surreal numbers as sign sequences, we give the sign sequence formula for logatomic numbers. In doing so, we relate logatomic numbers to fixedpoints of certain surreal functions.
The class of surreal numbers of J. H. Conway [7] is an inductively defined ordered field with additional structure. Conway uses the abstract notion of Dedekind inspired cut as a fundamental constructor to define numbers by wellfounded induction.
Indeed, any number is obtained from sets of previously defined numbers, as the “simplest” number with and . This relation is denoted . Conversely, any sets of surreal numbers with give rise to a unique simplest number . Thus the definition of comes with several features: an inductively defined order , an ordersaturation property, a corresponding ordinal rank called the birthday which represents the minimal ordinal number of inductive steps required to yield . For instance has birthday whereas has birthday and has birthday .
The versatility of this construction allowed Conway and several authors after him to define a rich structure on . In particular, he defined ring operations that are compatible with the ordering and turn into an ordered field extension of the reals, as well as an extension of the ordered semiring of ordinal numbers under Hessenberg sum and product.
Conway also discovered that enjoys a natural structure of field of Hahn series as per [10]. Every surreal number can be expressed as a possibly transfinite sum of additively irreducible numbers called monomials. Moreover, monomials can be parametrized by a morphism , called the map, for which Conway gave a an equation using the cut presentation of numbers. Whereas the latter presentation is useful to produce inductive definitions of functions, by induction on the birthday for instance, the presentation of numbers as Hahn series is useful to compute certain operations on . In particular, it can be used to describe H. Gonshor's exponential function [9], and A. Berarducci and V. Mantova's surreal derivation [5].
The sign sequence presentation of surreal numbers, invented and studied by H. Gonshor in [9], is a way to give precise description of surreal numbers. In this picture, numbers are sequences of signs indexed by ordinals, or equivalently, nodes in the binary tree . The birthdate measure then coincides with the domain, called length, of the sign sequences. A natural relation of simplicity, which corresponds to the inclusion of sign sequences in one another and is denoted , emerges as a more precise measure of the complexity of numbers.
Sign sequences are by no means an optimal tool to describe surreal numbers in the context of the ordered field, exponential ordered field, Hahn series field, or differential field structures. For instance given numbers with known sign sequences, computing the sign sequence of , or even formulating the properties of that of in general are open problems. However, sign sequences behave relatively well with respect to operations that preserve simplicity under certain conditions. This includes the map, the transfinite summation which identifies numbers with Hahn series, and the exponentiation of monomials. In this context, sign sequences can be useful tools to understand the behavior of length under the operations, as well as deriving general properties of certain classes of numbers with respect to the relations and .
Describing the sign sequences of elements in a class often requires a reference with respect to which they can be given. One way to give such a reference is to find a parametrization of , i.e. a bijection for a certain subclass , and then describe the sign sequence of in terms of that of for each . This presumes that the behavior of function on sign sequences may be understood, suggesting that should be compatible in a sense with the relations and . There enters the notion of surreal substructure of [3]. Surreal substructures are subclasses of that are isomorphic to under the restrictions of and to . The isomorphism then being unique, one may rely on it to relate the sign sequence of with that of .
Surreal substructures naturally appear when defining certain operations on , see for instance [3, Sections 6 and 7]. The study of surreal substructure yields tools to express and compute sign sequences. In particular, expressing a structure using classes of fixed points for given parametrizations yields a method to compute the sign sequence formula of . Relying extensively on [3], we will develop the relevant notions in Sections 1 and 2.
A surreal substructure of particular interest is the class of logatomic numbers. Those are numbers such that the fold iteration of of the logarithm at yields a monomial for each . This structure plays a crucial role in the definition of derivations on that are compatible with the exponential and the structure of field of series of . It is also used in the investigation of the properties of expansions of numbers as transseries. This class was characterized by Berarducci and Mantova [5, Section 5] who defined such a derivation and proved a fundamental structure property for . Finally plays a role in the definition of the first surreal hyperexponential function [4]. Our our goal in this article is to compute the sign sequence formula for . This will in particular give the sign sequence for for each surreal number whose length is strictly below the first number .
In this task, we are continuing work of S. Kuhlmann and M. Matusinski in [13]. Indeed, they considered a surreal substructure properly contained in and determined its sign sequence formula. The relation between and was subject to the conjecture
which turned out to be false [5, Proposition 5.24]. The correct relation between and was later found by Mantova and Matusinski [14]. We will rely on their sign sequence formula and a presentation of using classes of fixed points to derive our formula.
As Gonshor, we define numbers as sign sequences.
Given a surreal number , we extend its sign sequence with for all . Given and , we also introduce the restriction to as being the initial segment of of length , i.e. for and for .
The ordering on is lexicographical: given distinct elements , there exists a smallest ordinal with and we set if and only if .
For , we say that is simpler than , and write , if . The partially ordered class is wellfounded. We write if and . For , we write
for the set of numbers that are strictly simpler than . The set is wellordered with order type . Moreover, is the union of the sets
Every linearly ordered subset of has a supremum in . Indeed, we have , and for all with . 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.
Notice that is a proper class. For instance, the linearly ordered class is embedded into the partial order through the map . We will thus identify ordinals numbers as surreal numbers.
Any nonarchimedean ordered field admits a nontrivial valuation called the natural valuation [1, Section 3.5]. The natural valuation of is related to its archimedean class, i.e. the class
In NBG, the corresponding valued ordered field can be embedded [11] into a field of generalized Hahn series, as defined by Hahn [10]. The existence of such embeddings, called Kaplansky embeddings, usually requires the axiom of choice.
Having defined the structure of ordered field, Conway noticed that for , the instances where choice is required could be circumvented by the use of the inductive nature of . Indeed, in , each archimedean class has a unique minimal element denoted , and the class is a subgroup of which can be used to carry the definition of the Kaplansky embedding.
Moreover, Conway defined a parametrization of . This is a strictly increasing morphism which also preserves simplicity:
The operation coincides with the ordinal exponentiation with basis .
For ordinals , we will denote their ordinal sum, product, and exponentiation by , and . Here, we introduce the operations and of [3, Section 3.2] on . Those are natural extensions of ordinal arithmetic to , which we will use in order to describe sign sequences.
For numbers , we write for the number 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)) 
The operation clearly extends ordinal sum. We have for all .
We write for the number of length whose sign sequence is defined by
(α<ℓ(y),β<ℓ(x)) 
The operation extends ordinal product. Special cases can be described informally: given and , the number is the concatenation of with itself to the right “many times”, whereas is the number obtained from by replacing each sign by itself many times. We have and for all .
The operations also enjoy the properties which extend that of their ordinal counterparts as is illustrated hereafter. We refer to [3, Section 3.2] for more details.
If is an ordinal and is a function , then uniquely determines a surreal number defined inductively by the rules
,
for ,
for limit ordinals .
We say that is a sign sequence formula for , which we identify with the informal expression
where ranges in . For instance, the ordered pair is a sign sequence formula of . In general, we look for such formulas where alternates between and , that is, where ranges in and where, for each ordinal with , we have . It is easy to see that every surreal number admits a unique such alternating sign sequence formula. We refer to this formula as the sign sequence formula of said number.
If is a function, we may look for a function whose value at each number is a sign sequence formula of . We then consider as a sign sequence formula for .
As an example, we now state Gonshor's results regarding the sign sequences of monomials. For , we write for the order type of , that is the order type when is seen as a sign sequence, or equivalently the ordinal number of signs in the sign sequence of . We have if and only if . Gonshor found the following sign sequence formula for monomials:
For instance, we have
Let us also specify a consequence that we will use often in what follows.
where ranges in . Proposition 4 yields . On the other hand, we have where is defined as the transfinite concatenation
where ranges in . For , we have and , so
Example
For , the class has parametrization . We have
For , the class has parametrization . We have
The class of monomials has parametrization .
Let be a surreal substructure and let be a convex subclass of . This means that for with , we have . Then we have a simple criterion to decide whether is a surreal substructure:
In particular, nonempty open intervals , where , are surreal substructures. The following proposition gives a sign sequence formula for in certain particular cases.
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 . Given , we write for the fold imbrication of into itself. We have .
Example
Less elementary results include the relation of [3, Proposition 7.3].
The relation of imbrication is related to the inclusion of surreal substructures in the following way:
If are surreal substructures with , then the class is that of numbers with .
Notice that may not itself be a surreal substructure in general. We thus consider the following notion of closure:
In general, when is a surreal substructure, we write .
Example
For , the structure is closed, with .
For , the structure is closed, with where .
The structure is closed, and the parametrization of is usually denoted , since it extends the ordinal function which parametrizes the fixed points of .
The structure (where ) is closed. Moreover, its structure of fixed points coincides with the class of fixed points of the function , although this last one is not a surreal isomorphism. Informal expansions
of numbers in this class can be replaced by the formulation . The sign sequence of can be computed in terms of that of using Proposition 19 below.
Let be a closed surreal substructure and let denote its alternating sign sequence formula. This formula is “continuous” or “closed” in the sense that for any limit number , we have and . If for each surreal number and for we write for the nonzero surreal number defined by , we then have
where ranges in . Thus it is enough to compute the numbers to determine sign sequences of element of . In the case in Gonshor's formulas for the map and the map and KuhlmannMatusinski's formula for the map, those numbers are ordinals or opposites of ordinals. If is not closed, then for every limit number for which there will be an additional term such that is not always zero, and that we have, for all ,
where . Such terms must then be computed independently. Thus it may be a good first step towards computing a sign sequence formula for to check whether it is closed or not, and where closure defects occur.
Example
By [9, Chapter 9], the sign sequence formula for is given by the parameters:
Assume again that is a closed surreal substructure and let . Our goal in this paragraph is to compute provided we know on . The identity suggests that may be a form of limit of as tends to infinity. Indeed, we claim that one can always find such that is a chain with supremum . More precisely, we have:
Assume that is a limit number. Notice that is the simplest element of with , whereas is the simplest number with . We deduce that .
Otherwise, there is a number and a sign with . Let
In this section, we define the exponential function of [9, Chapter 10] as well as the surreal substructures and .
Inductive equationsRecall that by Conway's construction, or in Gonshor's setting by [9, Theorem 2.1], given sets of numbers with , there is a unique minimal, or simplest, number with
Let be a surreal substructure. Carrying the previous property through the parametrization of , we obtain, given a surreal substructure and subsets with , a unique minimal element of with
In order to write certain equations, we will write, for , and . Notice that we have by definition. Thus if is another surreal substructure, the surreal isomorphism sends onto
(1) 
Surreal exponentiation Gonshor uses an inductive equation to define the exponential function. Given and , set
Let be such that is defined on . Notice that for instance, for and , we should have
whereas similar inequalities hold for . This led Gonshor to define
Gonshor proved that this equation is warranted and that it does define a strictly increasing bijective morphism . We write for the reciprocal of .
The functions and The function interacts with the map in the following way:
More precisely, for every strictly positive number and every number , we have
where the strictly increasing and bijective function and its reciprocal have the following equations in and [9, Theorems 10.11 and 10.12]:
The function was entirely studied by Gonshor who gave formal results such as the characterization of its fixed points and as well as a somewhat informal description of the sign sequence of for any strictly positive number given that of . We will recover part of his results in a different approach in Section 4.
The class of logatomic numbers is defined as the class of numbers with for all . In other words, we have the equality
The class was first described by Berarducci and Mantova [5] in order to define a derivation on . We next describe the parametrization of .
Consider, for and the function
Recall that the class of monomials is that of numbers which are simplest in each class
Similarly, Berarducci and Mantova proved that logatomic numbers are exactly the simplest numbers of each class
Moreover, they derived [5, Definition 5.12 and Corollary 5.17] the following equation for the parametrization of :
where respectively range in , , and .
Moreover, the following formula [2, Proposition 2.5] is know for :
(3) 
In [4], we defined a strictly increasing bijection which satisfies the equation
(4) 
Morever, this function is surrealanalytic at every point in the sense of [6, Definition 7.8] and satisfies for all . This function can be seen as a surreal counterpart to Kneser's transexponential function [12].
In order to define , we relied on a surreal substructure of socalled truncated numbers. They can be characterized as numbers with . The function is defined on the class of truncated series by the equation:
This equation realizes a strictly increasing bijection .
Now consider the set of surreal numbers with length . By [8, Corollary 5.5], the structure is an elementary extension of the real exponential field. Moreover, for , there is with . It follows that each element of is truncated, and that for , the sets and are mutually cofinal with respect to one another. Thus on , the equation for becomes
We claim that for all . We prove this by induction on . Let such that the result holds on . Noticing that is initial in , we get
So and coincide on .
The structure of numbers was introduced first and studied in detail by S. Kuhlmann and M. Matusinski in [13]. It was designed as an intermediate subclass between fundamental monomials and logatomic numbers. The relation between and is given by the imbrication of [14, p 21].
Similarly to monomials and logatomic numbers, numbers can be characterized as simplest elements in each convex class
The parametrization of is given by the equation
Moreover, by [3, Corollary 13 and Theorem 6.16], the structure is closed.
In order to compute the sign sequence of numbers, Kuhlmann and Matusinski rely on an intermediate surreal substructure denoted . This surreal substructure is defined by the imbrication relation: . Indeed, since , the structure exists and is unique.
For we let denote the number with if and if . We also extend the functions and to with and . For , we define and . The parametrization of admits the following sign sequence formula
Our goal is to compute the sign sequence of in terms of that of , for all numbers . Recall that , so there is a surreal substructure with . We write for all . We have , whence . The computation in [13] of sign sequences of numbers in can thus be used to derive part of the result.
For where and , we have . By [13, Theorem 4.3], we have
Our goal is to extend this description to the sign sequences of numbers for all , relying on the known values for all and . More precisely, we will compute on all intervals
Since can be realized as the reunion
this will cover all cases. The sign sequence of can then be computed using Proposition 4 twice. In order to compute the sign sequence for , we first describe the action of and on sign sequences in certain cases.
Let be fixed and write .
Recall that we have so . By [13, Lemma 4.2], we have . We set
so if and if . We will treat the cases and in a uniform way.
We now fix and we set . By the previous lemma, we can write the interval as the surreal substructure
Thus is straightforward to compute in terms of sign sequences.
We have and . Thus the interval is subject to the computation given in Lemma 9(b). That is, we have:
if .
.
Moreover, for , we have and . This implies that and that . Since with , we deduce the following equation for on :
We see that is the surreal isomorphism , so .
For , we set
and
Recall that if , we have , so is subject to the computations of Lemma 9(a). We have , so is subject to the computations of Lemma 9(b), whence
if .
if .
if .
if .
if .
if .
Piecing the descriptions of Propositions 23 and 28 together, one obtains a full description of the sign sequence of for . We will only require part of those descriptions to reach our goal of describing sign sequences in . For , we set . We will characterize using fixed points of the surreal substructure
Notice that this structure depends only on . Both and are closed so is closed by Lemma 15, so the class is also closed.
For where and , we have
For where and , we have
First assume that . For and , the number is a monomial, which implies by Lemma 24 that has the form with . This implies in particular that .
Let such that the result holds at and consider and . There is a with . Since , we have . Now by Proposition 23(b). Let with . We have , so by Lemma 24, there is a number with and . Thus . By Proposition 4 and Lemma 5, we have
We identify
The inductive hypothesis applied at yields , so . We thus have
We next prove ). Since it implies that , this will yield ). We prove ) by induction on . Note that the result is immediate for . Let be such that the formula holds at , let and consider . Our inductive hypothesis is
Thus
This are the desired results. The formula follows by induction.
We next prove ). Let where , so that is a generic element of . By the same computations as above, we have
This proves the formula for . Now let be such that the formula holds at . Let with be a generic element of . We have so , thus Proposition 28(a) yields
If the set of numbers with is cofinal in , then by closure of , so .
We apply the results of the previous sections to give the sign sequence formula for .
As a consequence, we have .
Note that the functions , , and preserve nonempty suprema. Moreover, the identity is valid for . So by the previous lemma, we need only prove the claim at successors cases. Let , set , and define
We have . Let with (this is trivially the case for ). We have
It follows that . The second claim is valid since we have . Let , and for define . We have . Let with (this is trivially the case for ). We have
Our work on sign sequences is summed up in the following table where we distinguish the cases and .


In order to obtain the formula for , one need only use the relation and apply Proposition 4 twice.
Acknowledgments. This work is supported by the French Belgian Community through a F.R.I.A. grant.
class of surreal numbers 1
class of ordinal numbers 1
simplicity relation 1
th sign of 2
length of 2
is simpler than 2
set of strictly simpler numbers 2
supremum of the chain in 3
unique monomial in the archimedean class of 3
class of monomials 3
map, i.e. parametrization of 3
ordinal sum 3
ordinal product 3
ordinal exponentiation 3
sum concatenation 3
product concatenation 3
isomorphism 6
imbrication of into 6
class of fixed points of 6
parametrization of 7
exp Gonshor's exponential function 9
class of logatomic numbers 9
parametrization of 9
class of numbers 10
parametrization of 10
structure with 11
parametrization of 11
structure with 11
parametrization of 11
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