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vincent.bagayoko@lilo.org
Log-atomic 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 log-atomic numbers. In doing so, we relate log-atomic numbers to fixed-points 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
well-founded 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 order-saturation 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 semi-ring
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 log-atomic 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.
, where 
is an ordinal number. We call 
the 
and the map 
the .
We write 
for the class of surreal numbers.
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 well-founded. We write 
if and 
.
For , we write
for the set of numbers that are strictly simpler than . The set 
is
well-ordered 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 non-archimedean ordered field 
admits a
non-trivial 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.
which is simplest in the class 
,
i.e. a number of the form 
for a certain 
.
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.
,
we have
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:
be a number.
The sign sequence formula for 
is
For instance, we have
Let us also specify a consequence that we will use often in what follows.
be such that the maximal ordinal 
with 
is a limit. We have 
.
be maximal with
and write 
.
Since 
is strictly positive, we have 
, whence 
.
Let
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
.
of such that 
and 
are isomorphic. The isomorphism 
is unique, denoted and called the
parametrization of .
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:
of is a surreal substructure if and only if
every subset of has strict upper and lower
bounds in .
In particular, non-empty open intervals 
,
where 
, are surreal
substructures. The following proposition gives a sign sequence formula
for 
in certain particular cases.
be a non-zero ordinal. Let 
and be numbers.
if and only if 
and 
if
and only if 
so the given descriptions cover all
cases. We will only derive the formulas for the interval 
, the other ones being symmetric. The number
is the root of so for
, we have 
for a certain number 
. The
class 
is convex in and
for 
, we have 
. The function 
defined
on defines a surreal isomorphism, so we have
if .
On the other hand, we have
, which
yields the other part of the description.
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 , it follows from Lemma 3 that we have
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:
, we have 
if and only if there is a surreal substructure 
with 
.
be a surreal substructure. We say that a number is -fixed if 
. We write 
for the class of -fixed numbers.
If 
are surreal substructures with 
, then the class 
is
that of numbers with 
.
is a
surreal substructure, then 
.
Notice that may not itself be a surreal
substructure in general. We thus consider the following notion of
closure:
is closed if we
have 
for all non-empty set 
such that 
is linearly ordered.
and 
are closed surreal substructures, then so is 
.
is closed,
then 
is a closed surreal substructure.
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 non-zero 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
Kuhlmann-Matusinski'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 
for the 
-map are
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:
is a closed surreal substructure and let . We have
,
and set 
. Since is closed, so is 
,
which implies that 
is -fixed.
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
is 
, the sign of 
is as well, so 
.
It follows by induction that is a 
-chain whose supremum
lies in each 
for 
by
closure of those structures, so 
.
Since the sign of 
is , we have 
,
whence 
. Now
is the simplest -fixed number
such that the sign of is , whence 
.
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 log-atomic 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 log-atomic 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 surreal-analytic 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 so-called
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
log-atomic numbers. The relation between and is given by the imbrication
of [14, p 21].
Similarly to monomials and log-atomic 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 
.
and [13, Lemma 4.2]. The
second immediately follows.
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 have 
by [13, Theorem 4.3(1)]. By Lemma 20, we have 
so 
, so
. Since 
, we have 
.
It follows that 
.
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:
, 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 
.
.
, if the number 
is a monomial, then there is 
with
and 
.
where 
. Since 
is
a monomial with 
, by
Proposition 4, the sign sequence of
must start with 
, which
contradicts the hypothesis on .
We deduce that there is a number with 
. Since 
is
a monomial and by the same argument, the sign sequence of must start with at least 
many
signs 
, that is, we must have
.
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
, and 
,
we have
if 
.
if .
if .
if 
.
and we have
if 
.
if .
.
.
We have 
so 
.
We now compute
.
, and , we have
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
that 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
is true for all
.
, the structure is closed.
and 
are closed and by Lemma 15, the surreal
substructure is closed. Therefore is closed. By the previous theorem, the structure can be expressed as 
for a
certain number depending on
and . So by Lemma 15,
it is closed.
is closed.
be a non-empty
chain in . We want to prove
that 
is log-atomic. We may assume that has no -maximum.
If the set 
of 
-numbers
with 
is -cofinal in ,
then 
by closure of 
,
so 
.
is not -cofinal
in . Let 
be -minimal with 
. Let 
and notice that
. Let
with 
. For 
, the number 
is -maximal in
with 
. By [3,
Corollary 7.13 and Theorem 6.16], we have 
for
all , whence 
. The set 
is cofinal
and coinitial in 
for the order 
. So for each element
there is a unique integer 
with 
. The equations (5) and (6)
imply that if 
are elements of 
with 
, then we must have
. It follows that 
exists in 
. If
where ,
then 
. If 
, then for 
with 
and 
, we
have 
and 
.
The structure 
is closed, so by Corollary 30 and Lemma 15, the structure 
is closed. So 
is log-atomic.
We apply the results of the previous sections to give the sign sequence
formula for .
. We have 
, which is the transfinite concatenation
As a consequence, we have 
.
is a
direct consequence of the closure of .
The computation of this supremum is left to the reader.
define 
and 
. Let 
be the function
defined on by 
.
We have
is closed, we
may rely on Proposition 19. We need only prove that for
and ,
we have 
. We prove this by
induction on 
along with the claim that 
.
Note that the functions 
,
, 
and
preserve non-empty 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
. The second
claim is valid since we have 
.
This concludes the proof.
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 log-atomic 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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