> <\body> <\hide-preamble> >> \; |||<\author-affiliation> Département de Mathématique Université de Mons, Le Pentagone 20, Place du Parc B-7000 Mons, Belgique |>>||<\author-affiliation> CNRS, LIX Campus de l'École polytechnique 1, rue Honoré d'Estienne d'Orves Bâtiment Alan Turing, CS35003 91120 Palaiseau, France |>>> Surreal numbers form the ultimate extension of the field of real numbers with infinitely large and small quantities and in particular with all ordinal numbers. Hyperseries can be regarded as the ultimate formal device for representing regular growth rates at infinity. In this paper, we show that any surreal number can naturally be regarded as the value of a hyperseries at the first infinite ordinal >. This yields a remarkable correspondence between two types of infinities: numbers and growth rates. > At the end of the 19-th century, two theories emerged for computations with infinitely large quantities. The first one was due to duBois-Reymond, who developed a\Pcalculus of infinities\Q to deal with the growth rates of functions in one real variable at infinity. The second theory of \Pordinal numbers\Q was proposed by Cantor as away to count beyond the natural numbers and to describe the sizes of sets in his recently introduced set theory. Du Bois-Reymond's original theory was partly informal and not to the taste of Cantor, who misunderstood it. The theory was firmly grounded and further developed by Hausdorff and Hardy. Hausdorff formalized duBois-Reymond's \Porders of infinity\Q in Cantor's set-theoretic universe. Hardy focused on the computational aspects and introduced the differential field of : such a function is constructed from the real numbers and an indeterminate (that we think of as tending to infinity) using the field operations, exponentiation, and the logarithm. Subsequently, this led to the notion of a Hardy field. As to Cantor's theory of ordinal numbers, Conway proposed a dramatic generalization in the 1970s. Originally motivated by game theory, he introduced the proper class> of , which simultaneously contains the set > of all real numbers and the class > of all ordinals. This class comes with a natural ordering and arithmetic operations that turn > into a non-Archimedean real closed field. In particular, +\>, >, >>, >-3*\> are all surreal numbers, where > stands for the first infinite ordinal. Conway's original definition of surreal numbers is somewhat informal and draws inspiration from both Dedekind cuts and von Neumann's construction of the ordinals: <\quote-env> \PIf 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.\Q The notation\|\|}>> is called . Conway proposed to consider > as the number between and . Indeed, it turns out that one may define a partial ordering > on > with \a> for any number > with a\R>. This so-called has the additional property that any > can be written canonicallyas <\eqnarray*> ||\a|}>>>|>|>|\b\a,b\a|}>>>|>|>|\b\a,b\a|}>.>>>> One may regard \a> as the set of surreal numbers that were defined before when using Conway's recursive definition. Conway's bracket is uniquely determined by the simplicity relation > and . The ring operations on> are defined in a recursive way that is both very concise and intuitive: given \|x|}>> and \|y|}>>, we define <\eqnarray*> |>|\|\|}>>>||>||}>>>||>|\-x|}>>>||>|+y,x+y\|x+y,x+y|}>>>||>|*y+x*y-x*y,x*y+x*y-x*y\|x*y+x*y-x*y,x*y+x*y-x*y|}>>>|||\x,x\x,y\y,y\y|)>.>>>> It is quite amazing that these definitions coincide with the traditional definitions when and are real, but that they also work for the ordinal numbers and beyond. Subsequently, Gonshor also showed how to extend the real exponential function to > and this extension preserves all first order properties of . Simpler accounts and definitions of can be found in . The theory of Hardy fields focuses on the study of growth properties of germs of actual real differentiable functions at infinity. An analogue formal theory arose after the introduction of by Dahn and Göring and, independently, by Écalle>. Transseries are a natural generalization of the above definition of Hardy's logarithmico-exponential functions, by also allowing for infinite sums (modulo suitable precautions to ensure that such sums make sense). One example of a transseries is <\equation*> f=\+2*|x>+6*|x>+\>+\*log log x>->-++>+\. In particular, any transseries can be written as a generalized series \\>f>*\> with real coefficients >\\> and whose (trans)monomials \\> are exponentials of other (generally \Psimpler\Q) transseries. The support \\\f>\0|}>> of such a series should be in the sense that it should be well ordered for the opposite ordering of the natural ordering > on the group of transmonomials >. The precise definition of a transseries depends on further technical requirements on the allowed supports. But for all reasonable choices, \Pthe\Q resulting field > of transseries possesses a lot of closure properties: it is ordered and closed under derivation, composition, integration, and functional inversion; it also satisfies an intermediate value property for differential polynomials. It turns out that surreal numbers and transseries are similar in many respects: both> and > are real closed fields that are closed under exponentiation and taking logarithms of positive elements. Surreal numbers too can be represented uniquely as Hahn series \>a>*\> with real coefficients >\\> and monomials in a suitable multiplicative subgroup > of >>. Any transseries \> actually naturally induces a surreal number |)>\> by substituting > for and the map f|)>> is injective. But there are also differences. Most importantly, elements of > can be regarded as functions that can be derived and composed. Conversely, the surreal numbers > come equipped with the Conway bracket. In fact, it would be nice if any surreal number could naturally be regarded as the value |)>> of a unique transseries at >. Indeed, this would allow us to transport the functional structure of > to the surreal numbers. Conversely, we might equip the transseries with a Conway bracket and other exotic operations on the surrealnumbers. The second author conjectured the existence of such a correspondence between > and a suitably generalized field of the transseries16>; see also for a more recent account. Now we already observed that at least surreal numbers > can be written uniquely as |)>> for some transseries \>. Which numbers and what kind of functions do we miss? Since a perfect correspondence would induce a Conway bracket on >, it is instructive to consider subsets \> with R> and examine which natural growth orders might fit between and . One obvious problem with ordinary transseries is that there exists no transseries that grows faster than any iterated exponential ,\>,\>. Consequently, there exists no transseries \> with |)>=,\>,\>>,\\|\|}>>. A natural candidate for a function that grows faster than any iterated exponential is the first >>, which satisfies the functional equation <\equation*> E>=exp E>. It was shown by Kneser that this equation actually has a real analytic solution on>>. A natural hyperexponential >> on ,\>\\c\\|}>> was constructed more recently in. In particular, >|)>=,\>,\>>,\\|\|}>>. More generally, one can formally introduce the transfinite sequence >|)>\>> of of arbitrary strengths >, together with the sequence >|)>\>> of their functional inverses, called . Each >> with \>> satisfies the equation <\equation*> E>=E>>|)> and there again exist real analytic solutions to this equation. The function >>> does not satisfy any natural functional equation, but we have the following infinite product formula for the derivative of every hyperlogarithm>>: <\equation*> L>=\\>>>. We showed in how to define >> and >> for any \> and ,\>>. The traditional field > of transseries is not closed under hyperexponentials and hyperlogarithms, but it is possible to define generalized fields of that do enjoy this additional closure property. Hyperserial grow rates were studied from a formal point of view in. The first systematic construction of hyperserial fields of strength \\>> is due to Schmeling. In this paper, we will rely on the more recent constructions from that are fully general. In particular, the surreal numbers > form a hyperserial field in the sense of, when equipped with the hyperexponentials and hyperlogarithms from. A less obvious problematic cut R> in the field of transseries > arises by taking <\eqnarray*> ||,+\>,+\+\>>,\|}>>>|||,+\>,+\+\>>,\|}>.>>>> Here again, there exists no transseries \> with f\R>. This cut has actually a natural origin, since any \Ptame\Q solution of the functional equation <\equation> f=+\> lies in this cut. What is missing here is a suitable notion of \Pnested transseries\Q that encompasses expressions like <\equation> f=+\+\+\>>>. This type of cuts were first considered in2.7.1>. Subsequently, the second author and his former PhD student Schmeling developed an abstract notion of generalized fields of transseries that may contain nested transseries. However, it turns out that expressions like() are ambiguous: one may construct fields of transseries that contain arbitrarily large sets of pairwise distinct solutions to(). In order to investigate this ambiguity more closely, let us turn to the surreal numbers. The above cut R> induces a cut |)>\R|)>> in >. Nested transseries solutions to the functional equation() should then give rise to surreal numbers |)>> with |)>\f|)>\R|)>> and such that |)>->,log|)>->|)>-\>>,\> are all monomials in>. In8>, we showed that those numbers |)>> actually form a class > that is naturally parameterized by a surreal number (> forms a so-called ). Here we note that analogue results hold when replacing Gonshor's exponentiation by Conway's >map \\> (which generalizes Cantor's >-map when >>). This was already noted by Conway himself and further worked out by Lemire. of the present paper will be devoted to generalizing the result from 8> to nested hyperseries. Besides the two above types of superexponential and nested cuts, no other examples of \Pcuts that cannot be filled\Q come naturally to our mind. This led the second author to conjecture that there exists afield> of suitably generalized hyperseries in such that each surreal number can uniquely be represented as the value|)>> of a hyperseries\>> at >. In order to prove this conjecture, quite some machinery has been developed since: a systematic theory of surreal substructures, sufficiently general notions of hyperserial fields, and definitions of >|)>\>> on the surreals that give> the structure of a hyperserial field>. Now one characteristic property of generalized hyperseries in > should be that they can uniquely be described using suitable expressions that involve , real numbers, infinite summation, hyperlogarithms, hyperexponentials, and a way to disambiguate nested expansions. The main goal of this paper is to show that any surreal number can indeed be described uniquely by a hyperserial expression of this kind in >. This essentially solves the conjecture from by thinking of hyperseries in > as surreal numbers in which we replaced > by . Of course, it remains desirable to give a formal construction of > that does not involve surreal numbers and to specify the precise kind of properties that our \Psuitably generalized\Q hyperseries should possess. We intend to address this issue in a forthcoming paper. Other work in progress concerns the definition of a derivation and a composition on>. Now Berarducci and Mantova showed how to define a derivation on > that is compatible with infinite summation and exponentiation. In, it was shown that there actually exist many such derivations and that they all satisfy the same first order theory as the ordered differential field >. However, as pointed out in, Berarducci and Mantova's derivation does not obey the chain rule with respect to >>. The hyperserial derivation that we propose to construct should not have this deficiency and therefore be a better candidate for derivation on > with respect to >. In this paper, we will strongly rely on previous work from. The main results from these previous papers will be recalled in Sections, , and. For the sake of this introduction, we start with a few brief reminders. The field of > was defined and studied in. It is a field of Hahn series =\|]>|]>> in the sense of that is equipped with a logarithm >\\>>, a derivation :\\\>>, and a composition :\\\,\>\\>. Moreover, for each ordinal \>, it contains an element >> such that <\eqnarray*> |\f>||>>|+1>>\\>>>||+1>>-1>>|>>||\\>\+1>.>>>> for all \,\>> and all ordinals ,\>. Moreover, if the Cantor normal form of > is given by =\>*n> with \\\\>, then we have <\eqnarray*> >>||>>n>\\\\>>n>.>>>> The derivation and composition on > satisfy the usual rules of calculus and in particular a formal version of Taylor series expansions. In, Kaplan and the authors defined the concept of a to be afield =\|]>|]>> of Hahn series with a logarithm >\\>> and a composition law :\\\,\>\\>>, such that various natural compatibility requirements are satisfied. For every ordinal >, we then define the >> of strength > by >:\,\>\\,\>;\>\f>>. We showed in how to define bijective hyperlogarithms >:,\>\,\>> for which > has the structure of a hyperserial field. For every ordinal >, the functional inverse >:,\>\,\>> of >> is called the of strength >. The main aim of this paper is to show that any surreal number > is not just an abstract hyperseries in the sense of, but that we can regard it as a hyperseries in >. We will do this by constructing a suitable unambiguous description of in terms of >, the real numbers, infinite summation, the hyperexponentials, and the hyperlogarithms. If |)>> for some ordinary transseries , then the idea would be to expand as a linear combination of monomials, then to rewrite every monomial as an exponential of a transseries, and finally to recursively expand these new transseries. This process stops whenever we hit an iterated logarithm of >. In fact, this transserial expansion process works for any surreal number >. However, besides the iterated logarithms (and exponentials) of >, there exist other monomials \>\\\\\1|}>> such that |)>> is a monomial for all \>. Such monomials are said to be atomic>. More generally, given \>, we say that > is \>>>-atomic> if >|)>\> for all \\>>. We write >>> for the set of such numbers. If we wish to further expand an \>>>-atomic monomial > as a hyperseries, then it is natural to pick > such that > is not \+1>>>-atomic, to recursively expand L>>>>, and then to write =E>>>. Unfortunately, the above idea is slightly too simple to be useful. In order to expand monomials as hyperseries, we need something more technical. In Section, we show that every non-trivial monomial \\> has a unique expansion of exactly one of the two following forms: <\equation> \=\>*>|)>|)>>, where >\>, \>, and \>, with \log>|)>|)>>; or <\equation> \=\>*>>|)>|)>>, where >\>, \>, \,\\\>> with *\\\>, \log>>|)>|)>>, and where > u> lies in >\L\> *\>>. Moreover, if =1> then it is imposed that =0>, =1>, and that cannot be written as +\*\> where \>, \>, \>>, and\supp \>>. After expanding > in the above way, we may pursue with the recursive expansions of > and as hyperseries. Our next objective is to investigate the shape of the recursive expansions that arise by doing so. Indeed, already in the case of ordinary transseries, such recursive expansions may give rise to nested expansions like <\equation> >+\>+\>+\>>> One may wonder whether it is also possible to obtain expansions like <\equation> >+\>+\>+\ \>+log log log \>+log log \>+log \. Expansions of the forms() and() are said to be and , respectively. The axiom for fields of transseries in prohibits the existence of ill-nested expansions. It was shown in that > satisfies this axiom . The definition of hyperserial fields in does not contain a counterpart for the axiom. The main goal of section 4 is to generalize this property to hyperserial fields and prove the following theorem: <\theorem> Every surreal number is well-nested. Now there exist surreal numbers for which the above recursive expansion process leads to a nested expansion of the form(). In 8>, we proved that the class> of such numbers actually forms a . This means that ,\,\|)>> is isomorphic to ,\,\>|)>> for the restriction >> of > to >. In particular, although the nested expansion() is inherently ambiguous, elements in > are naturally parameterized by surreal numbersin>. The main goal of Section is to prove a hyperserial analogue of the result from8>. Now the expansion() can be described in terms of the sequence >,>,>,\>. More generally, in Section we the define the notion of a in order to describe arbitrary nested hyperserial expansions. Our main result is the following: <\theorem> Any nested sequence > induces a surreal substructure > of nested hyperseries. In Section, we reach the main goal of this paper, which is to uniquely describe any surreal number as a generalized hyperseries in >. This goal can be split up into two tasks. First of all, we need to specify the hyperserial expansion process that we informally described above and show that it indeed leads to a hyperserial expansion in >, for any surreal number. This will be done in Section, where we will use labeled trees in order to represent hyperserial expansions. Secondly, these trees may contain infinite branches (also called paths) that correspond to nested numbers in the sense of Section. By Theorem, any such nested number can uniquely be identified using a surreal parameter. By associating a surreal number to each infinite branch, this allows us to construct a unique in > for any surreal number and prove our main result: <\theorem> Every surreal number has a unique hyperserial description. Two numbers with the same hyperserial description are equal. Let ,\,1,\|)>> be a totally ordered (and possibly class-sized) abelian group. We say that \\> is if it contains no infinite ascending chain (equivalently, this means that > is well-ordered for the opposite ordering). We denote by |]>|]>>|]>|]>>|field of well-based series with real coefficients over >> the class of functions \\> whose support <\equation*> supp f\\\\f|)>\0|}>|support of a series>> is a . The elements of > are called monomials and the elements in >*\> are called . We also define <\equation*> term f\>*\\\\supp f|}>,|set of terms of a series>> and elements \term f> are called terms in . We see elements of > as formal >f>*\> where >\f|)>\\> for all \\>. If \>, then \max supp f\\>>|> is called the of. For \\>, we define \>\\\>f>*\>\>>|truncation \\>f>*\> of > and >\f1>>>>|1>>>. For \>, we sometimes write g>g>| and g>> if f>. We say that a series \> is a of and we write f>g>|g-f>> if \g>. The relation > is a well-founded partial order on > with minimum . By , the class > is field for the pointwise sum <\equation*> \>>+g>|)>*\, and the Cauchy product <\equation*> f*g\>*\=\>f>*g>|)>*\, where each sum *\=\>f>*g>> is finite. The class > is actually an ordered field, whose positive cone >=\\f\0|}>> is defined by <\equation*> >\\\f\0\f>\0|}>>>. The ordered group ,\,\|)>> is naturally embedded into >,\,\|)>>. The relations > and > on > extend to > byg>|>*\>> g>|r\\>,\r*>> <\eqnarray*> g>|>|>*\>>|g>|>||\r\\>|\>,\r*.>>>> We also write g>g>|g> and f>> whenever g> and f>. If are non-zero, then g> ( g>, g>) if and only if \\> ( \\>, =\>). \ We finally define>>|series \> with 1>> >>|series \> with 1>> ,\>>|series \> with 0> and 1>> <\eqnarray*> >>|>|\\supp f\\>|}>>>|>>|>|\\supp f\\>|}>=\\f\1|}>,and>>>|,\>>|>|\\f\\|}>=\\f\0\f\1|}>.>>>> Series in >>, >> and ,\>> are respectively called , , and . If |)>I>> is a family in >, then we say that |)>I>> is if\ <\enumerate-roman> I>supp f> is well-based, and I\\\supp f|}>> is finite for all \\>. Then we may define the sum I>f> of |)>I>> as the series <\equation*> I>f\>I>|)>>|)>*\. If =\|]>|]>> is another field of well-based series and :\\\> is >-linear, then we say that > is if for every well-based family |)>I>> in >, the family |)>|)>I>> in> is well-based, with <\equation*> \I>f|)>=I>\|)>. The field >>|field of logarithmic hyperseries> of plays an important role in the theory of hyperseries. Let us briefly recall its definition and its most prominent properties from . Let > be an ordinal. For each\\>, we introduce the formal hyperlogarithm >\L> x>> and define \>>\>>|group of logarithmic hypermonomials of force \>> to be the group of formal power products =\\>\>>>>> with >\\>. This group comes with a monomial ordering> that is defined by <\eqnarray*> \1>|>|\\\\>\0|}>>\0.>>>> By what precedes, \>\\\>|]>|]>>\>>|field of logarithmic hyperseries of force \>> is an ordered field of well-based series. If ,\> are ordinals with \\>, then we define ,\|)>>> to be the subgroup of \>> of monomials > with >=0> whenever \\>. As in , we define <\eqnarray*> ,\|)>>>|>|,\|)>>|]>|]>>>|>|>|\>\\>>>|>|>||]>|]>.>>>> We have natural inclusions ,\|)>>\\\>\\>, hence natural inclusions ,\|)>>\\\>\\>. The field \>> is equipped with a derivation :\\>\\\>> which satisfies the Leibniz rule and which is strongly linear: for each \\>, we first define the logarithmic derivative of >> by >>\\\>\>\\\>>. The derivative of a logarithmic hypermonomial \\\>> is next defined by <\eqnarray*> \>|>|\\>\>*\>>|)>*\.>>>> Finally, this definition extends to \>> by strong linearity. Note that \>=\\>\>>> for all \\>. For \\>> and \>, we will sometimes write >\\ f>. Assume that =\>> for a certain ordinal >. Then the field \>> is also equipped with acomposition :\\>\\\>,\>\\\>> that satisfies: <\itemize-dot> For \\>,\>>, the map \>\\\>;f\f\g> is a strongly linear embedding 6.6>. For \\>> and \\>,\>>, we have h\\\>,\>> and h|)>=g|)>\h> 7.14>. For \\>,\>> and successor ordinals \\>, we have >>\\>>=\>>-1> 5.6>. The same properties hold for the composition :\\\,\>\\>, when > is replaced by >. For \\>, the map \>\\\>;f\f\\>> is injective, with range ,\|)>>> 5.11>. For \,\|)>>>, we define > \>>> \>>|unique series in > with > \>|)>\\>>> to be the unique series in \>> with> \>\\>=g>. Following , we define > as the class of sequences <\equation*> a:\\ of \Psigns\Q indexed by arbitrary ordinals \>>|class of ordinals>. We will write > for the domain of such a sequence and |]>\> for its value at \dom a>. Given sign sequences and , we define>|simplicity relation> <\eqnarray*> b>|>|dom b\\\dom a,a|]>=b|]>|)>>>>> Conway showed how to define an ordering, an addition, and a multiplication on > that give > the structure of a real closed field. See 2> for more details about the interaction between > and the ordered field structure of >. By 21>, there is a natural isomorphism between> and the ordered field of well-based series |]>|]>>, where > is a certain subgroup of>,\,\|)>>. We will identify those two fields and thus regard > as a field of well-based series with monomials in >. The partial order ,\|)>> contains an isomorphic copy of ,\|)>> obtained by identifying each ordinal > with the constant sequence \\>> of length >. We will write \> to specify that > is either an ordinal or the class of ordinals. The ordinal >, seen as asurreal number, is the simplest element, or >-minimum, of the class ,\>>. For ,\\>, we write \\>\\>|ordinal sum of > and >> and \\>\>|ordinal product of > and >> for the non-commutative ordinal sum and product of > and >, as defined by Cantor. The surreal sum and product +\> and *\> coincide with the commutative Hessenberg sum and product of ordinals. In general, we therefore have +\\\\\> and *\\\\\>. For \>, we write >>>>|ordinal exponentiation with base > at >> for the ordinal exponentiation of base > at >. Gonshor also defined an exponential function on > with range >>. One should not confuse >> with *log \|)>>, which yields a different number, in general. We define <\equation*> \>\>\\\|}>, Recall that every ordinal > has a unique Cantor normal form <\equation*> \=\>*n+\+\>*n, where \>, ,\,n\\0>> and ,\,\\> with \\\\>. The ordinals > are called the of > and the integers > its . We write \\>\\>|\\>> for each exponent > of >> ( \\>\\>|\\>> for each exponent > of >>) if each exponent > of the Cantor normal form of > satisfies \\>> ( \\>>). If ,\> are ordinals, then we write \\>\\>|*n\\> for all \>> if *\\\>, we write \\>\\>|n\\,\\\*n>> if there exists an\> with \\*n>, and we write \\>\\>|\\\\>> if both \\> and \\> hold. The relation > is a quasi-order on>>>. For ,\,\\> with \\>> and \\>>, we have +\=\\\>. In particular, we have +1=\\1> for all \>. If \> is a successor, then we define > to be the unique ordinal with =\+1>.>|=\+1> if > is a successor and =\> if > is a limit> We also define \\> if > is a limit. Similarly, if =\>>, then we write >\\>>>>|>> for =\>>>. >> We already noted that Gonshor constructed an exponential and a logarithm on > and>>, respectively. We defined hyperexponential and hyperlogarithmic functions of all strengths on ,\>> in. In fact, we showed1.1> how to construct acomposition law :\\,\>\>>|composition law :\\,\>\>> with the following properties: <\description-compact> >>For \>, \,\>> and ,\>>, we have a\,\>> and <\equation*> f\a|)>=g|)>\a. >>For ,\>>, the function \;f\f\a> is a strongly linear field morphism. >>For \>, ,\>> and \> with \a>, we have <\equation*> f\|)>=\>>\a|k!>*\. >>For \> and ,\>> with b>, we have >\a\\>\b>. >>For \> and ,\>>, there is ,\>> with >\b>. Note that the composition law on > also satisfies to (but not >), with each occurrence of > being replaced by >. For \>, we write >> for the function ,\>\,\>;a\\>\a>, called the >. By and >, this is a strictly increasing bijection. We sometimes write > a\L>> for ,\>>. We write >> for the functional inverse of >>, called the >. For ,\> with \\>, the relation +\>=\>\\>> in >, combined with , yields <\equation> |\a\,\>|\>,L+\> a=L> L> a, For \>, the relation +1>>\\>>=\+1>>-1> in >, combined with , yields <\equation> |\a\,\>|\>,L+1>>>>|)>=L+1>>-1, and we call this relation the for +1>>>. Let >> and write \a>> for the coefficient in > in the Hahn series representation of . There is a unique infinitesimal number > with *\*|)>>. We write >> for the natural logarithm on >\>. The function defined by <\equation> log a\L|)>+log> r+\>|k+1>*\, is called the on >>. This is a strictly increasing morphism >,+|)>\,+|)>>> which extends >. It also coincides with the logarithm on >> that was defined by Gonshor. Given \>, we write >>> for the class of numbers ,\>> with > a\>> for all \\>>. Those numbers are said to be \>>>\>>>-atomic number> and they play an important role in this paper. Note that =>> and <\equation*> L>> +1>>>=+1>>> for all \>, in view of (). There is a unique >>-atomic number 6.20>, which is the simplest positive infinite number >.\ Each hyperlogarithmic function >>> with \0> is essentially determined by its restriction to >>>, through a generalization of. More precisely, for ,\>>, there exist\\>> and \>>> with \L>-L>|)>\L>>. Moreover, the family >>\>>|)>>\L>|)>|)>*\|)>\>>>> is well-based, and the hyperlogarithm >>> is given by <\equation> L>>=L>>|)>+\>>>> \>|)>>\L>|)>|k!>*\. <\definition> 6.10>> We say that \,\>> is >> if \1>>, if > is positive and purely large. For \\>, we say that \,\>> is >>truncated>>>>-truncated series> if <\equation*> |\\\supp \>|\>,|\\\\>|\>,\\\>>> \>\\. If >>|)>> is defined, then > is >>-truncated if and only if \>>>|)>|)>>>, for all \\>>. Given =\>> with \>, we write ,\>>,\>>|class of >-truncated series> for the class of >-truncated numbers. Note that ,1>=>\,\>>. We will sometimes write >|)>\E>>> when \,\>>. For ,\>>, there is a unique >-maximal truncation >> of which is >-truncated. By 7.17>, the classes <\equation> a+>\b=a\\\\,b\\>> \>\|)>|}> with ,\>> form a partition of ,\>> into convex subclasses. Moreover, the series >>>>|>-maximal >-truncated truncation of > is both the unique >-truncated element and the >-minimum of the convex class containing . We have <\equation*> ,\>=L> > by 7.6>. This allows us to define a map >:,\>\>> by >\E>>> b|)>>>>>. In other words, <\equation*> E>>>>=\>>|)>, for all ,\>> (see also 7.23>). The formulas and admit hyperexponential analogues. For all ,\>>, there is a \\> with \a-\>\>|\>>\E>>>>. For any such >, there is a family ,k>|)>\>\\\>>> with =\>> such that ,k>\E>>>|)>*\|)>\>> is well-based and <\equation> E> a=E>\>,k>\E>>>|k!>*\|)>. See 7.1> for more details on ,k>|)>\>>. The number > E>>>> is a monomial with > E>>>\,1>\E>>>|)>*\\,2>\E>>>|)>*\\\>, so <\equation> L> E>>>\L> E> a. In , we introduced the notion of . A surreal substructure is a subclass > of > such that ,\,\|)>> and ,\,\|)>> are isomorphic. The isomorphism \\>> is unique and denoted by>>. For the study of > as a hyperserial field, many important subclasses of > turn out to be surreal substructures. In particular, given =\>\>>, it is known that the following classes are surreal substructures: <\itemize-dot> The classes >>, ,\>> and >> of positive, positive infinite and infinitesimal numbers. The classes > and >> of monomials and infinite monomials. The classes >> and >>> of purely infinite and positive purely infinite numbers. The class >> of \>>-atomic numbers. The class ,\>> of >-truncated numbers. We will prove in Section that certain classes of nested numbers also form surreal substructures. Given a subclass > of > and >, we define <\equation*> ||>>|>|\b\a\b\a|}>>|>|>|>|>>>|>>|>|\b\a\b\a|}>>||>|>|>>>|>>>|>|>\a>>||>>|>|>>>>>>> If > is a subclass of > and are subsets of > with S>, then the class <\eqnarray*> >>|>|\l\L,l\a|)>\r\R,a\r|)>|}>>>>> is called a in >. If >> contains a unique simplest element, then we denote this element by>> and say that > is a (of >>) in >. These notations naturally extend to the case when > and > are subclasses of > with \\>. A surreal substructure > may be characterized as a subclass of > such that for all cut representations > in >, the cut >> has a unique simplest element 4.7>.\ Let > be a surreal substructure. Note that we have >\|a>|}>> for all >. Let > and let>> be a cut representation of in >. Then > is >,a>|)>> in the sense that has no strict upper bound in >> and has no strict lower bound in >>4.11(b)>. Let \> be a subclass, let > be a surreal substructure and \\> be a function. Let ,\> be functions defined for cut representations in > such that ,\> are subsets of > whenever > is a cut representation in >. We say that ,\|)>> is a for if for all >, we have <\equation*> \>,a>|)>\\>,a>|)>,F=>,a>|)>\|\>,a>|)>|}>>. Elements in >,a>|)>> (resp. >,a>|)>>) are called ( ) of this cut equation at. We say that the cut equation is if <\equation*> \\\,F>|)>=\|\|}>> for all cut representations > in >. For instance, given \>, consider the translation :\;a\a+r> on >. By 3.2>, we have the following uniform cut equation for > on >: <\equation> |\a\|\>,a+r=+r,a+r\|a+r,a+r|}>. Let \> with \0> and set \\>>. We have the following uniform cut equations for >> on >> and >> on ,\>> 8.1>: <\eqnarray*> |\\\>|\>,L> \>||> \>>\|L> \>>,L\> \|}>,\>>>>|||> L> \>>\|\> L> \>>,L\> \|}>.>>||\\\,\>|\>,E>>>|||E>> \>|)>,E>,\>>>>E>,\>>>>|>>>>>||||E\> \,\> E>,\>>>\> E>,\>>>|>.>>>> where <\eqnarray*> >>|||>|if > is a limit>>>|>*n\n\\|}>>|if =\+1> is a successor.>>>>>>>>>> A > on a surreal substructure > is a set-sized group of strictly increasing bijections \\> under functional composition. We see elements of > as actions on> and sometimes write and for \> rather than g> and>. We also write > for the functional inverse of \>. Given such a function group >, the collection of classes>|class of numbers with g,h\\,g a\b\h a>> <\equation*> \\\\\f,g\\,f a\b\g a|}> with \> forms a partition of > into convex subclasses. For subclasses \>, we write |]>\\>\>. An element > is said to be >>-simple element> if it is the simplest element inside >. We write >>>>|class of >-simple elements> for the class of >-simple elements. Given \>, we also define >>>>|projection \>>> to be the unique >-simple element of>. \ The function >> is a non-decreasing projection of ,\|)>> onto >,\|)>>. The main purpose of function groups is to define surreal substructures: <\proposition> 6.7 and Proposition6.8>> The class >> is a surreal substructure. We have the uniform cut equation <\equation> |\z\|\>,\>> z=*\>> z\ \>> z|}>>. Note that for >>, we have b> if and only if a\\ b>. We have the following criterion to identify the >-simple elements inside >. <\proposition> 6.5>> An element of > is >-simple if and only if there is a cut representation > of in > with L\a\\ R>. Equivalently, the number \> is >-simple if and only if a>\a\\ a>>. Given be sets of strictly increasing bijections \\>, we define>|comparison between sets of strictly increasing bijections> Y>| and are mutually pointwise cofinal> <\eqnarray*> Y>||def>>||\a\\|\>,|\f\X|\>,|\g\Y|\>,f a\g a>>|Y>||def>>|YY\X>>|Y>||def>>||\a\\|\>,|\f\X|\>,|\g\Y|\>,f a\g a>>|Y>||def>>||\a\\|\>,|\f\X|\>,|\g\Y|\>,f a\g a.>>>> If Y>, then we say that is with respect to . For >, we also write Y> or g> instead of \Y> and >.\ Given a function group > on >, we define a partial order > on > by g\\>. We will frequently rely on the elementary fact that this ordering is compatible with the group structure in the sense that <\equation*> |\|f,g,h|\>\\|\>,g\id>\f*g*h\f*h. Given a set of strictly increasing bijections \>, we define |X|\>> to be the smallest function group on > that is generated by , |X|\>\\\\f\n\\,f,\,f\X\X|}>>|X|\>>|function group generated by >. The examples of surreal substructures from the beginning of this section can all be obtained as classes >> of >-simplest elements for suitable function groups > that act on >, >>, or ,\>>, as we will describe now. Given \> and \>>, we firstdefine>|translation a+r>> >|homothety s*a>> >|power function a>> <\equation*> |||>|>|a+c>|acting on > or ,\>>>>>|>|>|r*a>|acting on >> or ,\>>>>>|>|>|a>|acting on >> or ,\>>.>>>>>> For =\>\>, we then have the following function groups>|function group \r\\|}>>> >|function group \s\\>|}>>> >|function group \s\\>|}>>> >|function group |E H L:n\\,s\\>|\>>> >>|function group ,L\n\\|}>>> <\eqnarray*> >|>|\c\\|}>>>|>|>|\r\\>|}>>>|>|>|\r\\>|}>>>|>>|>||E> H L>:\\\,r\\>|\>>>|>>|>|> \> E>.>>>> Now the action of > on > yields the surreal substructure >\>> as class >simplest> elements. All examples from the beginning of this section can be obtained in a similar way: <\itemize-dot> The action of > on > (resp. ,\>>) yields >> (resp. >>>). The action of > on >> (resp. ,\>>) yields > (resp. >>). The action of > on ,\>> yields \=E>>>. The action of >> on ,\>> yields >>. The action of >> on ,\>> yields ,\>>. We have <\eqnarray*> >>>>||>>>|>>>>||>.>>>> Let \>=>\\\\|}>> and \>=>\\\\|}>>. We will need a few inequalities from . The first one is immediate by definition and the fact that \E>>. The others are 6.9, Lemma6.11, and 6.17>>, in that order: <\eqnarray*> *\>>|>|>>>|\>>|>|> H L>>>||E>\\\\|\>>|>|>if > is a limit>>>||\\\\\\|\>,|\r,s\1|\>,E> H L>>|>|> H L>.>>>> From (), we also deduce that <\equation> > H L>\\\\,r\\|}>\\>. In this section, we prove Theorem, i.e. that each number is well-nested. In Section we start with the definition and study of hyperserial expansions. We pursue with the study of paths and well-nestedness in Section. The general idea behind our proof of Theorem is as follows. Assume for contradiction that there exists a number that is not well-nested and choose a simplest (i.e. >minimal) such number. By definition, contains a so-called \Pbad path\Q. For the ill-nested number from(), that would be the sequence <\equation*> \>+\>+\ \>+log log log \>+log log \>,\>+\ \>+log log log \>,\ \>,\ From this sequence, we next construct a \Psimpler\Q number like <\equation*> a\>+\>+\>+\ \>+log log log \>> that still contains a bad path <\equation*> \>+\>+\ \>+log log log \>>,\>+\ \>+log log log \>,\ \>,\, thereby contradicting the minimality assumption on . In order to make this idea work, we first need a series of \Pdeconstruction lemmas\Q that allow us to affirm that > is indeed simpler than ; these lemmas will be listed in Section. We will also need a generalization > of the relation > that was used by Berarducci and Mantova to prove the well-nestedness of > as a field of transseries; this will be the subject of Section. We prove Theorem in Section. Unfortunately, the relation > does not have all the nice properties of >. For this reason, Sections and are quite technical. Recall that any number can be written as a well-based series. In order to represent numbers as hyperseries, it therefore suffices to devise a means to represent the infinitely large monomials> in>>. We do this by taking a hyperlogarithm > \> of the monomial and then recursively applying the same procedure for the monomials in this new series. This procedure stops when we encounter a monomial in > \>. Technically speaking, instead of directly applying a hyperlogarithm >> to the monomial, it turns out to be necessary to first decompose > as a product =\>*\> and write > as a hyperexponential (or more generally as the hyperlogarithm of a hyperexponential). This naturally leads to the introduction of of monomials \1>>, aswe will detail now. <\definition> We say that a purely infinite number \>> is > if =\\\*\>, for certain \>>, \>, and \>>. <\definition> Let \1>>. Assume that there are \>>, \>, \\\>>, \> and > such that <\equation> \=\>*> E>|)>>, with \L+1> E>>. Then we say that )> is a > if <\itemize> *\\\>; >\>\L\> *\>>; =1\=0 is not tail-atomic>|)>>. We say that )> is a > if =0> and >, so that >=\> and <\equation> \=\>*> \|)>>. Formally speaking, hyperserial expansions can be represented by tuples,\,\,\,u|)>>>. By convention, we also consider <\equation*> 1=\* E 0|)>, to be a hyperserial expansion of the monomial =1>; this expansion is represented by the tuple>>. <\example> We will give a hyperserial expansion for the monomial <\equation*> \=exp> \->+L+1> \|)>, and show how it can be expressed as a hyperseries. Note that <\equation*> u\log \=2*E> \->+L+1> \ is tail-atomic since > \> is log-atomic. Now > \=L> \> is a hyperserial expansion of type II and we have +1> \\E> \,>>. Hence =\> \->>*> \>|)>> is a hyperserial expansion. Let \> \>->>, so =\>*> \|)>>. We may further expand each monomial in >. We clearly have > \\>>. We claim that > \\>\L\> >>. Indeed, if we could write > \=L L*m>*\> for some \>> and \>>, then =L> L*m>*\|)>=L*> \-n> and *> \> would both be monomials, which cannot be. Note that > \=E>> E> \|)>=E>> \+1>>, so > \=E>> \+1>>> is a hyperserial expansion of type I. We also have >=exp*log \|)>> where *log \> is tailexpanded. Thus >=E*log \>> is a hyperserial expansion. Note finally that =L \> is a hyperserial expansion. We thus have the following \Precursive\Q expansion of >: <\equation> \=\>>>> \+1>-E*L \>>*> \|)>. <\lemma> Any \> has a hyperserial expansion. <\proof> We first prove the result for \>>, by induction with respect to the simplicity relation >. The >-minimal element of >> is >, which satisfies() for =\=0> and=1>>. Consider \>\|}>> such that the result holds on >>>>. By 6.20>, the monomial > is not >>-atomic. So there is a maximal \\>> with \>>, and we have \\> by our hypothesis. If there is no ordinal \\> such that >>\*\>>, then we have \>\L\> *\>>. So setting =\>, =0> and > \>, we are done. Otherwise, let \\> be such that \E>>\*\>>. We cannot have =0> by definition of >. So there is aunique ordinal > and aunique natural number \>> such that =\+\>*n> and\\>>>. Note that\\+1>>>. We must have =\+1>>: otherwise, +1>> \=L+\+1>>|)>+n> where +1>> \> and +\+1>> \> are monomials. We deduce that =0> and =\>*n>. Note that > \\L> \>>, \\*\>, and \*\>>, so =\*\>|)>>. We deduce that \\>. The induction hypothesis yields a hyperserial expansion =\>*> E>|)>>>. Since > is log-atomic, we must have =0> and =1>. If =L> \>, then \\>=\>>, since \*\>>. Thus =L> \=L+\> \> is a hyperexponential expansion of type II. If =L> E>>, then likewise \\>> and thus =L+\> E>> is a hyperexponential expansion of type I. This completes the inductive proof. Now let \>\>> and set \log \>. If > is tail-atomic, then there are \>>, \> and \>> with =\\\*\>. Applying the previous arguments to >, we obtain elements \\,\,u> with =L> E>> and *\\\>, or an ordinal > with =L> \>. Then =\>*> E>|)>>> or =\>*> \|)>>> is a hyperserial expansion. If > is not tail-atomic, then we have =E>> is a hyperserial expansion of type I. <\lemma> Consider a hyperserial expansion =L> E>>. Let \0> and define \\-1> if > is a successor ordinal and \\> if > is a limit ordinal. Let <\eqnarray*> >|>|+\where>>>|>|>|\>>>\\>>and>>>|>|>|\>>>\\>>.>>>> <\enumerate-alpha> Then > is \>>>-atomic if and only if =0> and either \\>> or =0>. If \\>>, then >>|)>=L> E>>. <\proof> We first prove ). Assume that > is \>>>-atomic. Assume for contradiction that \0> and let >*m> denote the least non-zero term in the Cantor normal form of >. Since \\>>>, we have +1>\\>> so +1>> \> is a monomial. But +1>> \=L\>>> E>-m> where \>>> E>> is a monomial: a contradiction. So =0>. If =0> then we are done. Otherwise >\*\>>, so we must have *\\\>>, whence \\>>. Conversely, assume that \\>> or =0>, and that =0>. If \0>, then then for all \\>>, we have > \=L+\> E>> where +\\\>, so > \> is a monomial, whence \>>>. If =0>, then for all \\>>, we have > \=L+\> \\>, whence \>>>. This proves ). Now assume that \\>>. So > E>> is \>>>-atomic by ). If =0> then we conclude that =L> E>=\>>|)>>. If \0>, then let >*m> denote the least non-zero term in its Cantor normal form. We have +1>\\>> and +1>> \=L+1>> L> E>-m\L+1>> L> E>>, so > E>=\>>|)>>. <\corollary> Let \>>, \\>>, \\>, and \*\>>. If > \\>\*\>>, then > is a successor ordinal and =\>*n> for some \>>. <\proof> Since > \\>\*\>>, we must have \0>. By Lemma, we have a hyperserial expansion =\>*> E>|)>>>. Since > is log-atomic, we have =\\\*L+1> E>\>, whence =0> and =1>. So =L> E>>. We have \*\>> so by (), we have \\>>. It follows that > \=L+\> E>> is a hyperserial expansion. But then +\> E>\>>> and (>) imply that \\>>. The condition that \\> now gives \\>, whence > is a successor and =\>*n> for a certain \>>, as claimed. <\lemma> Any \> has a unique hyperserial expansion (that we will call hyperserial expansion, henceforth). <\proof> Consider a monomial \1> with <\equation*> \=\>*> \|)>>, where \>>, \>, ,\\\>>, \>>, *\\\>, and \L+1> \>. Assume for contradiction that we can write =\>*> E> u|)>>> as a hyperserial expansion of type I with\\>. Note in particular that \1>, so +1> \> is log-atomic. We have <\equation*> log \=\\\ L+1> \=\\\*L+1> E>>. If =1>, then =0>, =0>, =1>, and > is not tail-atomic. But \\ L+1> \=u>, where +1> \\>>, so > is tail-atomic: a contradiction. Hence \1>. Note that L+1> \> and *L+1> E>>> are both the least term of >. It follows that =\>, =\>, and <\equation> L> \=L> E>>. Since *\\\>, we have <\equation*> E>>=\>> E>>|)>=\>> \|)>. Now >>\*\>>, so >> \|)>\\> and thus *\\\>. In particular \\>. Taking hyperexponentials on both sides of , we may assume without loss of generality that=0>> or that the least exponents > and > in the Cantor normal forms of > > differ. If =0>, then we decompose \\>*n> where \>> and \\>>. Since > \=E>>\>\*\>>, applying (>) twice (for >=\> and >=\*\>) gives +1>\\> and +1>\\*\>>, whence =\+1>>. But then >>=L>*n> \>, where \L> \\>*\> by (>). So >>\L\> *\>>: a contradiction. Assume now that \0>. (>) yields both > \\+1>>\+2>>>> and > E>>\+1>>\+2>>>, which contradicts . Taking =\\> and \max*\,\*\|)>>, this proves that no two hyperserial expansions of distinct types I and II can be equal. Taking =E>> with \\>, this proves that no two hyperserial expansions >*> E>|)>>,\>*> E>>|)>>> of type I with \\> can be equal. The two remaining cases are hyperserial expansions of type II and hyperserial expansions >*> E>|)>>> and >*> E>>|)>>> of type I with =\>. Consider a monomial \>> with the hyperserial expansions =\>*> \|)>>=\>*> \|)>>> of type II. As above we have =\>, =\>, and > \=L> \>. We deduce that =\>, so the expansions coincide. Finally, consider a monomial \1> with two hyperserial expansions of type I \; <\equation> \=\>*> E>|)>>=\>*> E>>|)>>. If =1>, then we have =\=0> and =\=0> and =\=1>, whence >, so we are done. Assume now that \1>. Taking logarithms in (), we see that =\>, =\>, and <\equation> L> E>=L> E>>. We may assume without loss of generality that\\>. Assume for contradiction that \\>. Taking hyperexponentials on both sides of , we may assume without loss of generality that=0>> or that the least exponents > and > in the Cantor normal forms of>> differ. On the one hand, (>) yields > E>\+1>>\+2>>>. Note in particular that > E>\>>, since *\\\>. On the other hand, if \0>, then (>) yields > E>>\>>\+1>>>; if =0>, then > E>>\>>. Thus is absurd: a contradiction. We conclude that =\>. Finally >=E>>> yields >, so the expansions are identical. <\lemma> If =\>*> E>|)>>> is a hyperserial expansion of type I, then we have <\equation*> supp \\supp u=\. <\proof> Assume for contradiction that \supp \\supp u>. In particular \L+1> E>>. Since 0>, there is \>> with r*\>, so +1> E>\\>: a contradiction. Let > be an ordinal with \\\> and note that 1\\\\\\\i\\=\|)>> for all\>. Consider a sequence>|value =P> of the path at 1+>> >|dominant monomial of >> >|constant coefficient of >> <\equation*> P=|)>\>=|)>\>=*\|)>\>in>*|)>>>>. We say that is a if there exist sequences |)>1\\>>, |)>1\\>>, |)>\>>, |)>\>>, and |)>1\\>> with <\itemize-dot> =\> and =0>; \term \> or \term u> for all \>; \\>\|}>\\=i+1> for all \>; For \>, the hyperserial expansion of > is <\eqnarray*> >||>*> E>>|)>>.>>>> We call > the of and we write \\>>|length of a path >*|)>>>>. We say that is if =\> and otherwise. We set \a>. For i\>, we define <\eqnarray*> ,a|)>>|>||)>>|if \supp \>>>>||)>>|if \supp u>>.>>>>>>>>> By Lemma, those cases are mutually exclusive so ,a|)>> is well-defined. For>>, we say that is a path in if \term a>. For >, we let k>> denote the path of length -k> in > with <\equation*> \i\-k,\k>,i>\\. <\example> Let us find all the paths in the monomial > of Example. We have a representation () of > as a hyperseries <\equation*> \=\>>> \+1>-E*L \>>*> \|)> which by Lemma is unique. There are nine paths in >, namely <\itemize-dot> one path |)>> of length ; three paths ,2*E>>> \+1>|)>>, ,-E*L \>|)>>, and ,\|)>> of length ; three paths ,2*E>>> \+1>,L> \|)>>, ,2*E>>> \+1>,1|)>> and \,-E*L \>,*L \|)>> of length; two paths ,2*E>>> \+1>,L> \,\|)>> and ,-E*L \>,*L \,\|)>> of length . Note that the paths which cannot be extended into strictly longer paths are those whose last value is a real number or >. Infinite paths occur in so-called nested numbers that will be studied in more detail inSection. <\definition> Let > and let be a path in a. We say that an index > is > for> if one of the following conditions is satisfied <\enumerate-numeric> > is not the >-minimum of >; =min supp u> and \0>; =min supp u> and =0> and \>; =min supp u> and =0> and \> and \supp \>. The index is > for > if it is not bad for >. If is infinite, then we say that it is > if|)>> is good for all but a finite number of indices. In the opposite case, we say that is a > path. An element > is said to be > every path in is good. <\remark> The above definition extends the former definitions of paths in. More precisely, a path with with =1> (whence =0>) for all >, corresponds to a path for these former definitions. The validity of the axiom for > means that those paths are good. With Theorem, we will extend this result to all paths. <\lemma> For \> \|)>1>> and for any path in >, we have \2>. For \\\> and for any path in , we have \3>. <\proof> Let \\\> and let be apath in \\>. If there is an ordinal > with =\>>, then the hyperserial expansion of \\> is > \>, so =1> if =0> and =2> otherwise. If there is an ordinal > with =\>>, then the hyperserial expansion of \\> is > \|)>> and =2>. Assume now that \\>1>>. If \\> is not tail-atomic, then hyperserial expansion of \\> is \\=\\\>>. If \\> is tail-atomic, then the hyperserial expansion of \\> is \\=\\\>*\\|)>>> for a certain log-atomic \\>. In both cases, 1>> is a path in some monomial in > \>, whence 1>|\|>\2> and \3>, by the previous argument. <\definition> Let be paths. We say that is a > , or equivalently that > , if there exists a > with k>>. For >, we say that is a > if there is a path in such that is a subpath of . We say that > if there is a subpath of which is a subpath in . Let be a finite path and let be a path with \supp u>\supp \>>. Then we define Q>Q>|concatenation of paths> to be the path ,\,P|)>,Q,\|)>> of length +>. <\lemma> Let \\>> and \>>. Let be a path in > with \1>. Then 1>> is a subpath in > \>. <\proof> By Lemma, we have \> \|)>1>>. If > has a hyperserial expansion of the form =\>*> \|)>>>, then 1>> must be a path in >. So > is non-zero and thus =1>. It follows that 1>> is a path in =\\\*> \|)>>>. Otherwise, let =\>*> E>|)>>> be the hyperserial expansion of >. If 1>> is a path in >, then it is a path in > as above. Otherwise, it is a path in . Assume that =1>. If =1>, then we have =0> and =\*u> so 1>> is a path in >. If \1>, then =\\\*L+1> E>> where +1> E>> is a hyperserial expansion, so 1>> is a path in >. Assume now that \1>, so =0>, =1>, and \\>. We must have \\>> so there are \> and \> with \\>> and =\+\>*n>. We have > \=L+\> E>-n> where +\> E>> is a hyperserial expansion, so 1>> is a path in> \>. <\lemma> Let ,\>>, \\>> and \>>. If is a path in >> with \2>, then 1>> is a subpath in*k> a>>. <\proof> We prove this by induction on *k>, for any number ,\>>. We consider ,\>>>, and a fixed path in >> with \1>. Assume that=k=1>. We have =a>> and =\>>>. Assume that >=\\\*\> for certain \>>, \>, and \>>. Let =L> E>> be the hyperserial expansion of >. If =\>, then =0> and the hyperserial expansion of >> is >=E>>. Therefore1>> is asubpath in =\>*>|)>>>. If \\>, then the hyperserial expansion of >> is >=L+1> E>>>. Therefore 1>> is a subpath in =\>*+1> E>|)>>>. Finally, if >>> is not tail-atomic, then1>> is a subpath in =*a>>>|)>>>, where \> is the sign of >>. Now assume that =1>, 1>, and that the result holds strictly below . We have a=E> where 1>> is a subpath in > by the previous argument. We have \\> for a certain \>>, so |)>\P1>> is a path in >. The induction hypothesis on implies that 1>=P1>> is a subpath in a>>. Assume now that \\> and that the result holds strictly below >. Write \>>. By(), there exist \>, \>, and \> with \\>\\> and <\equation*> E> a=E*n>*n> E>\\|)>. Assume for contradiction that there is a \> with >=L> \>. We must have \\>>, so there are a number \> and an ordinal \\> with =\+\>*n>. We have +\> \-n>>. By Lemma, this contradicts the fact that \2>. So by Lemma, there exist \\>> and \> with \\>>, *\\\>, >=L> E>>, and >\>\L\> *\>>. Since >\>>, we must have \\>> so there are a number \> and an ordinal \\> with =\+\>*n> (note that whenever >=\>). Thus > L+\>*n> E>+n=L+\> E>> is a monomial with hyperserial expansion +\> E>>. There is no path in of length 1>, so must be a path in +\> E>>. We deduce that 1>> is a path in . Consequently, > E>|)>\P1>> is a path in >> with =\1>. Applying times Lemma, we deduce that 1>=P1>> is a subpath in *n> E>>, hence in >*n> E> a|)>>. Consider a path in >*n> E> a|)>> with 1>=Ri>> for a certain 0>. Applying the induction hypothesis for*n> E> a> and*n> in the roles of and *k>, the path 1>> is a subpath in*n>*n> E> a|)>>=\> a>>. Therefore 1>> is a subpath in> a>>. We deduce as in the case =1> that 1>> is a subpath in*k> a>>. <\lemma> Let \>>, and \>> with \log \>. Let be a path in> with \1>. Then 1>> is a subpath in >*\>. <\proof> Let =\>*> E>|)>>> be a hyperserial expansion. The condition \log \> implies +\=\\\>, whence >*\=\\\>*> E>|)>>> is also a hyperserial expansion. In particular 1>> is a subpath in >*\>. <\corollary> Let =\>\>, \> with \\>, and \,\>>. If is an infinite path, then shares a subpath with > if and only if it shares a subpath with > E>>>. <\proof> Write \ =\>*m+\+\>*m> in Cantor normal form, with \\\\> and ,\,m\\>>> and let\ <\equation*> \\L>*m+\+\>*m> E>> for all ,k>. \ Assume that shares a subpath with >. In other words, there is a path in > which has a common subpath with . The path must be infinite, so by , it shares a subpath with >>=\>. Let us prove by induction on ,k> that shares a subpath with >>=\>. Assuming that this holds for k>, we note that > is \>*\>>-atomic, hence \>>>atomic. So shares a subpath with > by and the induction hypothesis. We conclude by induction that shares a subpath with =L> E>>>. Suppose conversely that shares a subpath with > E>>=\>. By induction on ,1>>, it follows from that shares a subpath with >. Applying to =E>>>, we conclude that shares a subpath with >. In this subsection, we list several results on the interaction between the simplicity relation > and various operations in ,+,\,>|)>\>|)>>. <\lemma> 3.3>> For >, we have <\equation*> a\b\*a\b. <\lemma> 5.12(a)>> For \> and \>>, we have <\equation*> sign*\\r*\. <\lemma> 4.20>> Let \>. For ,\> with ,\\supp \>, we have <\eqnarray*> \\\\\\>|>|\\.>>>> <\lemma> 4.21>> For ,\\>, we have <\eqnarray*> \\>|>|\\.>>>> <\lemma> 4.23>> Given ,a,b> in >> with supp \>, we have <\eqnarray*> \\>|>|\a>\\\b>.>>>> <\lemma> 4.24>> Given ,\\>> with \\>, we have <\eqnarray*> \\>|>|>\\>.>>>> <\lemma> Let \>> and \>>, let ,\\>\supp \>> with \\>|]>>, and let \>> with \supp \>. Then <\equation*> \\\\\\sign*\>\\\r*\\\>. <\proof> The condition \\>|]>> yields \\>. We have >\\>> by Lemma. The identity >>=>> implies that >\\*\>>, whence *\>\\>> by Lemma. Consequently, \sign*\>\\\r*\>>, by Lemma. Since =1\\>\>>, we may apply Lemma to \r*\> and \r*\\\> to obtain \r*\>\\\r*\\\>>. We conclude using the transitivity of >. <\lemma> Let \\>> with \1>. For ,\\,\>> with > E\> \\\>, we have <\equation*> \\\\E>>\E>>. <\proof> By(), we have <\equation*> E> \=|E\> \,\> E>,\>>>\> E>,\>>>|>. Since \\>, we have ,\>>\\,\>>> and ,\>>\\,\>>>, whence <\equation*> \> E>,\>>>\E> \\\> E>,\>>> Furthermore, we have > E\> \\\>, so\> \\E>>>. We conclude that >>\E>>>. In 8>, the authors prove the well-nestedness axiom > for > by relying on a well-founded partial order >>|Berarducci and Mantova's nested truncation relation> that is defined by induction. This relation has the additional property that <\equation*> |\|a,b|\>\>|\>,a\b\a\b. In this subsection, we define a similar relation > on > that will be instrumental in deriving results on the structure of ,>|)>\\>>|)>>. However, this relation does b\a\b> for all >. Given >, we define <\eqnarray*> b>||def>>||\n\\|\>,a\b,>>>> where |)>\>> is a sequence of relations that are defined by induction on , as follows. For , we set b>, if b> or if there exist decompositions <\eqnarray*> ||\sign*\>>|||\r*\\\,>>>> with \>> and \>. Assuming that > has been defined, we set b> if we are in one of the two following configurations: <\description-compact> We may decompose and as <\eqnarray*> ||\sign*\>*>|)>>>>|||\r*\>*> E>|)>>\\,>>>> where \>>, \>>, \\>>, *\\\>, \>, ,\>>,\ <\equation*> supp \\log E>,L+1> E>, and v>. If =1>, then we also require that =0>. We may decompose and as <\eqnarray*> ||\sign*\>>>|||\r*\>*\>\\,>>>> where \>>, ,\\>>, \>, \>,\\>, \log \>, and \\>. <\warning> Taking =1> in the first configuration, we see that > extends >. However, the relation > is neither transitive nor anti-symmetric. Furthermore, as we already noted above, we do have a,b\,a\b\a\b>. <\lemma> Let \\>>. Let ,\>> be numbers of the form <\eqnarray*> ||\r*\>>|||\s*\\\>>>> where ,\\>, \>> with =sign>, and ,\\>>. If \E> \> for sufficiently large \\>, then <\equation*> b\,\>\a\,\>. <\proof> Let \> and \\>>. Assume for contradiction that ,\>> and,\>>. Assume first that b>, so \>. Then \> E> b>>. Let \>> be such that k*\>\b>>. Since k*\>|)>\supp b>, we deduce that >|)>\\> E>>|)>>>, whence>>\,\>>. Modulo replacing by >>, it follow that we may assume without loss of generality that =k*\> for some \>> and some monomial >. On the one hand, is not >-truncated, so there are \|)>>> and > with \\\> andL>> \>|)>>. We may choose =\>*n> for certain \\> and \>>, soL>> \>*n>|)>>. On the other hand, > is >-truncated, so we have <\equation*> a+k*\\L>> \>*>|)>>|)>\L>> \>*n>|)>\a. We deduce that\L>> \>*>|)>>|)>-L>> \>*n>|)>>. If > is a successor, then choosing =\>>, we obtain \L>> \>*n>|)>+\>-L>> \>*n>|)>>>, so \1>: a contradiction. Otherwise, \ \\+1>,\|)>>\\> by , where +1>,\|)>>\+1>\\\\>\>>. Thus *\\+1>,\|)>>\\>>, whence *\\\+1>,\|)>>>: acontradiction. We now treat the general case. By a similar argument as above, we may assume without loss of generality that \s*\>. Assume that a>. Since is not >-truncated, there exists a \\> with \> E> a|)>\> E> b|)>>, whence \L> E> b>. But is >truncated, so \L\> E> b>. In particular \L> E> b>, so our hypothesis \L> \> implies that\L> L> E> b\L> E> b>: a contradiction. Assume now thata>. As in the first part of the proof, there are \\> and n\\> with \s*\\L>> \>*n>|)>> and >> \>*n>|)>\\\r*\>. Recall that \E> \> for sufficiently large \\>. Take \\> and \\> such that <\eqnarray*> >> \>*n>|)>>|>|>> \>*>|)>>>|>> \>|>|> is a limit.>>>>> Then L>> \>*>|)>-L>> \>*n>|)>>. If > is a successor, then choosing =\> yields 1>>, which contradicts the fact that > and > are infinitesimal. So > is a limit. Writing \max,\|)>>, we have \>. As in the first part of the proof, we obtain \+1>,\|)>>\\>>, so \\+1>,\|)>>\\\\>. In view of (), we also obtain \\>>, so \max,\|)>>: acontradiction. <\lemma> Let ,\\\>> with \\>. For ,\>>, we have <\eqnarray*> > u\\> v>|>|> E> u\\> E> v.>>>> <\proof> Assume that > u\\> v>. Let \>> and let > be its functional inverse in >>. We have \E> H L>> by (, ), whence E> H> L>>. Furthermore, E> \> v>, so <\equation> E> u\E> E> \> v. We want to prove that> u\> h E>|)> v>. By (), it is enough to prove that there is a\>> such that the inequality > E> g\E> h E>> holds on ,\>>.\ Assume that =\>. Setting H>\\>>, we have > h E>\g>, whence > g\h E>>, and > E> g\E> h E>>. Assume that \\>. We have > H>\H> so *2> H>\E> H\E> E>> by(). Thus> h\E*2> H> L>\E>>>. Consequently, > h E>\E> E>>, as claimed. If are numbers, then we write b|]>> for the interval ,max|]>>. <\proposition> For > with c> and c|]>>, any infinite path in shares asubpath with . <\proof> We prove this by induction on with c>. Let be an infinite path in . Assume that c>. If c>, then we have b> so is a path in . Otherwise, there are ,\\>, \>> and \> with \sign*\> and \r*\\\>. Then \s*\\t> for certain >, \>> and \> with \*\\r*\|]>>. We must have =\>. If is a path in >, then it is a path in . Otherwise, it is a path in *\>, so 1>> is a subpath in >, hence in . We now assume that c> where 0> and that the result holds for all ,b,c\> and n> with \c> and \\c|]>>. Assume first that > is in , and write <\equation*> ||||\sign*\>*>|)>>>>|||\r*\>*> E>|)>>\\>>>>>with>>u\v. Then we can write \s*\\t> like in the case when . If is a path in>, then it is a path in . So we may assume that is a path in*\>*>|)>>>. Note that we have\>*>|)>>\\>*> E>|)>>|]>>. Setting \*\>|)>>\>\L> E>|]>>, we observe that \supp \>, whence >*\>> is the hyperserial expansion of >. If 1>> is a path in >, then it is a path in >. Suppose that 1>> is not a path in >. Assume first that =1>>, so =0>, =0>, and is apath in |)>>>. Then Lemma implies that 1>> is asubpath in *u>, so 2>> is a subpath in. Otherwise, consider the hyperserial expansion >=L> E>>, >\>\L\> *\>> of >>. Since 1>> is not a path in >, it must be a path in . The number > E>> is \>>atomic, so we must have \\> and \\>>>. There are \> and \\>> such that =\+\>*n>>. Therefore +\> E>-n>. It follows by that 1>> shares a subpath with, whence so does . Let \>> \|)>>. Recall that \>\L> E>|]>>, so > \\L> L> E>|]>>. Now () implies that > E>\\>>|]>>, so > L> E>\L> \>>|]>=\>>. The function >=\>>>> is non-decreasing, so >> \|)>\\>> L> E>|)>|]>=v|]>>. But v>, so the induction hypothesis yields that 2>>, and thus , shares a subpath with . We deduce with Lemma that shares a subpath with>, hence with . Assume now that > is in , and write <\equation*> ||||\sign*\>>>|||\r*\>*\>\\>>>>>with>>\\\. Note that we also have \\\\*log \>. We may again assume that 1>> is a path in >. Write \s*\\t>, where \\>>, \>, and\>\\>*\>|]>\>. Then \\\\\*log \|]>> where \\\\*log \>. We deduce by induction that shares a subpath with>. By Lemma, it follows that shares asubpath with >, hence with . This concludes the proof. <\lemma> Let ,\\\>> and \> with *\\\>. Let ,\>> be of the form <\equation*> a=\\r*\>*> E>|)>>\\, with \>, \>>, \>>, ,\>>, \>, \> and > E>\supp \>. Consider an infinite path in ,\>> with b>. <\enumerate-roman> If >\supp \>, then shares a subpath with >. If >\supp \> and >*>|)>>\supp \>, then shares a subpath with >. If >\supp \> and >*>|)>>\supp \> and \\\sign*\>*>|)>>\,\>>, then shares a subpath with >. <\proof> \ . If >\supp \>, then we have \0>, so \1>. Let \supp \> with >\\>. Since >> and > are monomials, we have \log E>>, whence >\E>>. Our assumption that \supp \\log L> E>> also implies >\L> E>>. Hence >\>\L> E>|]>>. Now shares a subpath with >>, by Lemma. Since >\L> E>>, Proposition next implies that shares a subpath with >>. Using Lemma, we conclude that shares a subpath with >, and hence with >. . Let \supp \> with \\>*>|)>>>. It is enough to prove that shares a subpath with>. Since >, >*> E>|)>>>, and >*>|)>>> are monomials, we have >*> E>|)>>\\\\>*>|)>>>. Let \>*\|)>>>, so that \> E>\E>|]>>. In particular, we have \log \\1>. Moreover >\L> E>>, so using Lemma and Proposition, we deduce in the same way as above that shares a subpath with >. If \>>, then =\\\*log \>> is the hyperserial expansion of >, so shares a subpath with >. If \>>, then the hyperserial expansion of > must be of the form =E> E>>, since otherwise > would have at least two elements in its support. We deduce that shares a subpath with and that the hyperserial expansion of > is >*> E>|)>>>. Therefore shares a subpath with>. . We assume that > is not >-truncated whereas >\supp \> and >*>|)>>\supp \>. If =1>, then we must have >*>|)>>\1>, which means that \0> or that =0> and =-1>>. But then >*> E>|)>>\1>: a contradiction. Assume that \1>. By Lemma, we may assume without loss of generality that =0>. The assumption on > and the fact that ,\>> imply that > is non-zero. Write <\eqnarray*> >|>|>*>|)>>and>>>|>|>|>*> E>|)>>.>>>> So \r*\> and =\\sign*\>. Note that > must be infinitesimal since > is not >truncated. Thus > is also infinitesimal. By Lemma, we deduce that \> \\\>. We have >|)>\a>, so >|)>=\>, since and \a> are both >-truncated. Since > is not >truncated, there is an ordinal \\> with \> E>>|)>>. If \a>, then \\> E>|)>>, because is >-truncated. Thus \\> E>>|)>>. If \a>, then +\> E>>|)>\\>|]>\\>\a=\\r*\>, because > and are >-truncated. Now 0>, since \a>. We again deduce that\\> E>>|)>>. \ In both cases, we have > E>>\\\|]>> where \\>, so shares a subpath with> E>>>, by Proposition. It follows by that shares a subpath with>. We now prove that every number is well-nested. Throughout this subsection, will be an infinite path inside a number >. At the beginning of Section we have shown how to attach sequences |)>\>>, |)>\>>, to this path. In order to alleviate notations, we will abbreviate \r>, \\>, \u>, \\>, \\>, \\>, and \\> for all\>. We start with a technical lemma that will be used to show that the existence of a bad path in implies the existence of a bad path in a strictly simpler number than . <\lemma> Let >, let be an infinite path in and let \> such that every indexi> is good for >. For i>, let \|)>\>>, \r>, and \|)>\>>, so that |\,\,\|\>\> and <\eqnarray*> >||\\*\>*>>|)>>i|)>>>|>||\r*\>*>*E>>|)>>\\.>>>> Let \> and let \,\>> be a number with \u> and <\equation> c=\\\*sign|)>*\>*\>, for a certain \>> with \supp \>, \E>>> and \\>>>|]>> whenever =0>. For,0>, we define <\equation> c\\+\*\>*>>|)>> Assume that shares a subpath with >. If shares no subpath with any of the numbers ,\,\,\,\>, then we have\a>, and shares a subpath with >. <\proof> Using backward induction on , let us prove for ,0> that <\eqnarray*> |||||> c>|>|> u>>|>>>|>|>>|>*>>|)>>>|>|>>|>|>|>>| shares a subpath with >>>>||>|>|>|,\>>>|>|>|>>>> and that()> and()> also hold for . We first treat the case when . Note that \0> since it contains a subpath, so \,\>> or =1>. From our assumption that =\\\*sign|)>*\>*\>> and the fact that \\>>>|]>> if =0>, we deduce that \\>|]>>. Hence > c\\> u>> and()>. Note that ()> and ()> follow immediately from the other assumptions on>. If =0> then =\\u>. If =1>, then \L>*E>>>, since >*E>>\\>>>|]>> and \E>>\\>>>|]>>. Hence >\>*E>>|)>>> by and |)>*\>*\>\r*\>*>*E>>|)>>> by Lemmas and. Finally, \u> by Lemma, so ()> holds in general. Recall that is a subpath in>, but that it shares no subpath with > or>. In view of ()>, we deduce ()> from Lemma() and ()> from Lemma(). Combining ()>, ()> and ()> with the relation \u>, we finally obtain ()>. Let ,i-1|}>> and assume that (\U)>> hold for all \k>. We shall prove(\U)> if 0>, as well as ()> and()>. Recall that <\equation*> c=\+\*\>*>>|)>>. <\description-compact> )>>Recall that 0>. If \0> or \0>, then \\|]>> and(\U)> imply()>. Assume now that =\=0>. It follows since 0> that =1>>, so =E>>> and =E> u>. Since >>> is a hyperserial expansion, we must have \*\>>, so \\>. The result now follows from ()> and Lemma. )>>We know by ()> that shares a subpath with >. Since \,\>>, we deduce with that also shares a subpath with >>>, hence with >>|)>>>. In view of ()> and , we see that shares a subpath with >*>>|)>>> . Hence ()> gives that shares a subpath with >. )>> By()>, we have \u>. Now shares a subpath with > by ()>, but it shares no subpath with>. Lemma() therefore yields the desired result >>\supp \>. )>>As above, shares a subpath with >, but no subpath with>. We also have \u> and >>\supp \>, so ()> follows from Lemma(). )>>We obtain ()> by combining (\U)> and ()>. )>>The path shares a subpath with >, but no subpath with >. By what precedes, we also have >>\supp \> and >*>>|)>>\supp \>. Note finally that \,\>>. Hence \,\>>>, by applying Lemma() with >, >, >, >, and > in the roles of >, >, , , and . )>>It suffices to prove that >>\E>>>, since <\eqnarray*> ||>>\E>>>>||>|>>|)>>\>>|)>>)>>>||>|>*>>|)>>\\>*>>|)>>)>>>||>|*\>*>>|)>>\\*\>*>>|)>>>>||>|\\*\>*>>|)>>\\\\*\>*>>|)>>)>>>||>|\u.>>>> Assume that \1> and recall that <\eqnarray*> >||\\*\>>>|)>>>>|>||\\*\>>>|)>>.>>>> By Lemma, it suffices to prove that \u> and that > c\E>>> for all \\>. The first relation holds by ()>. By ()>, we have > c\\> u>. Therefore \E> *u\L\> E> u> by Lemma. This yields the result. Assume now that =1>. For ,i>, let <\eqnarray*> >|>|-\>>>|>|>|-\>.>>>> We will prove, by a second descending induction on ,k-1>, that the monomials > and > satisfy the premises of Lemma, ,\\1>, \\>|]>>, and \\>. It will then follow by Lemma that >\\>>, thus concluding the proof. If , then ,supp u\1>, because =1>. In particular ,\\1>>. Moreover, \\> follows from our assumption that \E>>>, the fact that >>\\>>>|]>\L> E>>>, and Lemmas and. If \0>, then we have \\>|]>> because \log \,log E>>>. Otherwise, we have =\\\>>>|]>=\>|]>>. Now assume that i>, that the result holds for , and that =1>. Again =1>> implies that ,\\1>. The relation \u> and Lemmas,, and imply that \\>. If \0>, then \\>|]>> by()>. Otherwise, we have =1>, because \,1>>. Since =1>, the number =\\\*E>>> is not tail-atomic, so we must have=1>. This entails that =\>> and =\>>. By the induction hypothesis at >, we have \\>|]>>. We deduce that\\>|]>>, so <\equation*> \\exp \>|]>=\>>|]>=\>|]>. It follows by induction that ()> is valid. This concludes our inductive proof. The lemma follows from ()> and ()>. We are now in a position to prove our first main theorem. <\render-proof|Proof of > Assume for contradiction that the theorem is false. Let be a >-minimal ill-nested number and let be a bad path in . Let \> be the smallest bad index in >. As in Lemma, we define \|)>\>>, \|)>\>>, and \r> for all i>. We may assume that 0>, otherwise the number \\\sign|)>*\>*>>|)>>> is ill-nested and \a>>: a contradiction. Assume for contradiction that there is a i> such that > or > is ill-nested. Set \0> if > is ill-nested and \1> otherwise. If =1>, then cannot share asubpath with>, so \\>> by Lemma, and \\*\>> is ill-nested. In general, it follows that \\\\*\*\>> is ill-nested. Let be a bad path in> and set \,\,P|)>\Q>. Then we may apply Lemma to , >, and > in the roles of , >, and . Since \u>, this yields an ill-nested number \a>: a contradiction. Therefore the numbers ,\,\,\,\> are well-nested. Since is bad for >, one of the four cases listed in Definition must occur. We set <\equation*> d\\sign|)>*\>>|>|() occurs>>>|\sign|)>*\>*>>|)>>>||.>>>>> By construction, we have \u>. Furthermore shares a subpath with >, so there exists a bad path in>. We have \,\>> by Lemma. If Definition() occurs, then we must have \0> so > is written as in () with > in the role of > and =\=1>. Otherwise, > is as in () for =E>>>. Setting \,\,P|)>\Q>, it follows that we may apply Lemma to > and> in the roles of > and . We conclude that there exists an ill-nested number \a>: a contradiction. In the previous section, we have examined the nature of infinite paths in surreal numbers and shown that they are ultimately \Pwell-behaved\Q. In this section, we work in the opposite direction and show how to construct surreal numbers that contain infinite paths of a specified kind. We follow the same method as in 8>. Let us briefly outline the main ideas. Our aim is to construct \Pnested numbers\Q that correspond to nested expressions like <\eqnarray*> ||>+\>-\>+\>-\>>>>>>>> Nested expressions of this kind will be presented through so-called >. Once we have fixed such a coding sequence >, numbers of the form() need to satisfy a sequence of natural inequalities: for any \> with 1>, we require that <\equation*> ||*>>|>||>|>>>|>+\*>>>|>||>|>+\>>>>|>+\>-\>>>>|>||>|>+\>-\*>>>>>|>+\>-\>+c*\>>>>>|>||>|>+\>-\>+c*\>>>>>>|||>||>>>> Numbers that satisfy these constraints are said to be . Under suitable conditions, the class of admissible numbers forms a convex surreal substructure. This will be detailed in Sections and, where we will also introduce suitable coordinates <\equation*> ||>||>+\>-\>+\>-\>>>>>||>|>||>-\>+\>-\>>>>||->|)>>>|>||>+\>-\>>>||>-a|)>>>|||>||>>>> for working with numbers in >. The notation() also suggests that each of the numbers ->>, >-a>, > should be a monomial. An admissible number > is said to be if this is indeed the case. The main result of this section is , that the class > of nested numbers forms a surreal substructure. In other words, the notation() is ambiguous, but can be disambiguated using a single surreal parameter. <\definition> Let \,\,\,\,\|)>\>\\\\\\>|)>>>. We say that > is a > if for all \>, we have <\enumerate-alpha> \>>; \,\>\>; =1|)>\=0\=0\\=1|)>|)>>; =\=0|)>\\\\\=\=1|)>>; j\i,\0\\\0|)>>. Taking =1> for all \>, we obtain a reformulation of the notion of coding sequences in8.1>. If =,\,\,\,\|)>\>> is a coding sequence and \>, then we write <\equation*> \k>\,\,\,\,\|)>\>, which is also a coding sequence. <\lemma> Let be an infinite path in a number > without any bad index for . Let \a\>> and \|)>\>> for all \>>. Then \,r,\,\,\|)>\>> is a coding sequence. <\proof> Let \>. We have \> because is a good index for >. We have \>>> and \,\>> by the definition of hyperserial expansions. If 0> and \0>, then we have \,\>> because \,\>> by the definition of paths. Lemma also yields \,\>>. This proves the conditions ) and ) for coding sequences. Assume that =1>. Then by the definition of hyperserial expansions, we have =0> and =a> is not tail-atomic. Assume that =0>. Then \1> so =1>>. We have =\\r*\> where \E>>> and > is not tail-atomic. This implies that > is not log-atomic, so =1>. Thus ) is valid. Assume that =\=0>. Recall that =r*>>|)>>=u\,\>>, so =\=1>. Since >>\*\>>, we have \*\>>, whence \\>. This proves ). Assume now for contradiction that there is an \\> with =\=0> for alli>>. By ), we have =\=1> for all i>, and the sequence |)>i>> is non-increasing, hence eventually constant. Let \i> with >=\> for all i>. For \>, we have >=E>*k> a+k>> so >\\>E>*k> >>=>*\>>. Therefore >>+1>>> is \+1>*\>>atomic: acontradiction. We deduce that ) holds as well. We next fix some notations. For all \> with j>, we define partial functions >,> and > on > by <\equation*> ||>|>|+\*\>*> a|)>>,>>|>|>|\\\\|)>,>>|>|>|.>>>>> The domains of these functions are assumed to be largest for which these expressions make sense. We also write <\equation*> >||>|>|i>\*\>>|>||>|>|k\j>\*\>>>>> We note that on their respective domains, the functions >, >, and > are strictly increasing if *\=1>, =1>, and =1>, respectively, and strictly decreasing in the contrary cases. We will write > and > for the partial inverses of > and >. We will also use the abbreviations <\equation*> >|>|>>|>|>|>>>>>\>|>|>>|>|>|>>>>> For all \>, we set <\equation*> ||||||||||||>>|>|-\*\>*supp \|)>>|>>|>|+\*\>*supp \|)>>>|>>|>|+\*\-\*\*\>*supp \>|)>>|>>|>|+\*\+\*\*\>*supp \>|)>>>|>>|>|>||\=0|\>>>||\*\=-1>>|> \|)>>|>>>>>>|>>|>|>||\=0|\>>>||\*\=1>>|> \|)>>|>>>>>>>|>|>|>\L>\L>>|>|>|>\R>\R>>>||>|\>L.>||>|\>R.>>>>> Note that <\eqnarray*> =0>|>|>=R>=\and>>>|=0>|>|>=R>=\.>>>> The following lemma generalizes8.1>. <\lemma> If R|)>>, then > is well defined for all \>. <\proof> Let us prove the lemma by induction on . The result clearly holds for. Assuming that > is well defined, let i> be minimal such that \0> or \0>. Note that we have \\\\\\>, so >\E>\\\E>=E>> where =\+\+\+\>. Applying> to the inequality <\equation*> L\a\R, we obtain <\equation*> \*|)>\\*a\\*|)>. Now if \0>, then <\eqnarray*> |)>>|>|+\*\>*>-\*\>*supp \|)>|)>>>>||)>>|>|+\*\>*>+\*\>*supp \|)>|)>>,>>>> whence <\equation*> \*\>*>-\*\>*supp \|)>|)>>\\*-\|\>\\*\>*>+\*\>*supp \|)>|)>>. Both in the cases when =1> and when =-1>, it follows that -\|)>/\*\>|)>>> is bounded from below by the hyperexponential >> of a number. Thus =L>-\|)>/*\>|)>|)>>|)>> is well defined and so is each > for k\j>. If =0>, then we have \0> and \; <\eqnarray*> |)>>|>|+\*\>*>-\*\*\>*supp \>|)>|)>>,>>||)>>|>|+\*\>*>+\*\*\>*supp \>|)>|)>>.>>>> Hence <\equation*> \*\>*>-\*\*\>*supp \>|)>|)>>\a-\\\*\>*>+\*\*\>*supp \>|)>|)>> Both in the cases when =1> and when =-1>, it follows that -\|)>/\*\>|)>>> is bounded from below by the hyperexponential >> of a number, so > is well defined and so is each > for k\j>. <\definition> Let \,\,\,\,\|)>\>> be a coding sequence and let >. We say that is >-> if > is well defined for all \> and <\eqnarray*> >||\\*\>*> a|)>>,>>|>|>|> a,>>|>|>|>|)>if \0>>.>>>> We say that > is > if there exists a >-admissible number. Note that we do not ask that >*> a|)>>> be a hyperserial expansion, nor even that > a> be a monomial. For the rest of the section, we fix a coding sequence =,\,\,\,\|)>>\>>. We write >>|class of admissible numbers> for the class of >-admissible numbers. If >, then the definition of > implicitly assumes that > is well defined for all \>. Note that if > is admissible, then so is k>> for \>. We denote by k>>k>>|class of k>>-admissible numbers> the corresponding class of k>>admissible numbers. The main result of this subsection is the following generalization of8.2>: <\proposition> We have =>. <\proof> Let \> and let \>. We have \,\>>. If =1>, then > is strictly increasing so we have <\eqnarray*> >\a\R>>|>|>|)>>\a\>|)>>>>||>|-\>*supp \\a\\+\>*supp \>>||>|-\\supp \>>||>|\a.>>>> If =-1>, then > is strictly decreasing and likewise we obtain \a\R\\\a>. We have *log E> a=\*-\|\*\>>|)>>. If =1>, then > is strictly increasing so we have <\eqnarray*> >\a\R>>|>|+\*\-\*\>*supp \>\a\\+\*\+\*\>*supp \>>>||>|>*supp \\log -\|\*\>>\\>*supp \>>||>|\log -\|\*\>>>>||>|> a\supp \.>>>> Likewise, we have >\a\R>\log E> a\supp \> if =-1>.\ Assume that \0> and =1>. If =1>, then we have \\>. Hence <\eqnarray*> >\L>\a\R>\R>>|>|> \\a\\\a>>||>|\\>|)>\\\a>>||>|\\>|)>.>>>> If =-1>, then we have \\>, whence <\eqnarray*> >\L>\a\R>\R>>|>|\\> \\\\a>>||>|>|)>\\\\\a>>||>|\\>|)>.>>>> Symmetric arguments apply when \0> and =-1>.\ We deduce by definition of > that =\>\|R|)>=>. As a consequence of this last proposition and 4.29(a)>, the class > is a surreal substructure if and only if > is admissible. <\example> Consider the coding sequence =,\,\,\,\|)>\>> where for all \>, we have <\eqnarray*> >||> \+L*2> \+L*3> \+\,>>|>||>|>||> \+L*2> \+L*3> \+\,>>|>||and>>>|>||.>>>> We use the notations from Section. We claim that > is admissible. Indeed for \>, set <\equation*> a\\+\>*>+\>*>>>>|)>>|)>. Given \> and j>, we have \a> and \R>. We deduce that R>, whence > is admissible. <\lemma> Let > and > be such that > and > have the same sign and the same dominant monomial. Then >. <\proof> For >>, we write y> if y> and and have the same sign. Let us prove by induction on \> that > is defined and that -\\b-\>. Since this implies that \b>, that -\|\*\>>\supp \>, and that \\>|)>> if 0>, this will yield>. \ The result follows from our hypothesis if . Assume now that -\\b-\> and let us prove that -\\b-\>. Let <\equation*> c\-\|\*\>>|)>. We have \-\|\*\>>|)>>=E> a\,\>>, so =L>|)>> is defined. Moreover \\>> a|]>> so \\>|]>>. Since \\>|)>=\>|)>>, we deduce that -\>\a-\>, whence in particular -\\a-\>. This concludes the proof. <\corollary> We have 1>=\>1>|]>>. <\proof> For 1>>, and \>>, we have \\>=\>> so \b-\>. We conclude with the previous lemma. <\lemma> For > and \>>, we have > a\\> b>. <\proof> Let i> be minimal with \0> or \0>. We thus have ,b\\\\*\>|]>> so \b>. We have =E+\+\> a> and =E+\+\> b> where \\\\\1>. We deduce by induction using Lemma that > a\\> b>. In this subsection, we assume that > is admissible. For \> we say that a k>>-admissible number is k>> if we have > a\>\L\> *\>> for all\>. We write k>>>|class of k>>-nested numbers> for the class of k>>-nested numbers. For we simply say that is >-nested and we write \0>>>|class of >-nested numbers>. <\definition> We say that > is > if for all \>, we have <\equation*> k>=\+\*\>*> k+1>|)>>. Note that the inclusion k>\\+\*\>*> k+1>|)>>> always holds. In 8.4>, we gave examples of nested and admissible non-nested sequences in the case of transseries, i.e. with =1> for all \>. We next give an example in the hyperserial case. <\example> We claim that the sequence > from Example is nested. Indeed, let \> and k+1>>. We have \\>*> b|)>> for a certain ,\>> withL>> \>. Let us check that the conditions of Definition are satisfied for \+\>*> a|)>>. First let \supp \>. We want to prove that \log E> a>. We have =L>*n> \> for a certain \>>. Now 2*L>> \>, so > a\E*2>>>> \>=L>>+2|)>\\>. Secondly, let \supp \>. We want to prove that \\>*> a|)>>. We have =L>*n> \> for a certain \>>. Then >*> a|)>\\>> by the previous paragraph. Now +\\3*L> \> so >\\> \>\\>. Finally, we claim that \\>>. This is immediate since the dominant term > of >*> b|)>> is positive infinite, so \\\\\\>>. Therefore > is nested. A crucial feature of nested sequences is that they are sufficient to describe nested expansions. This is the content of Theorem below. <\lemma> Let 1>>. If \1>, or =1> and >> is not tail-atomic, then the hyperserial expansion of > \>> is <\equation*> E> \>=E>>>> If =1>, >=\\\*\> is tail-atomic, and >=L> E>> is a hyperserial expansion, then \1>> and the hyperserial expansion of >> is <\equation*> exp b>=\>*> E>|)>>. <\proof> Recall that =b>>. By Corollary, we have >\1>>,. So we may assume without loss of generality that >>. We claim that >\>\L\> *\>>. Assume for contradiction that >\L\> *\>> and write >=L> \> accordingly. Then Corollary implies that =0>, in which case we define 0>, or =\+1>> for some ordinal > and =|)>>*n> for some \>>. Therefore >\*\>>, so \*\>>. This implies that <\equation*> b=\. Recall that \b>. Assume that , so =0>. Since is -atomic, we also have =0>. Let 1> be minimal with \0> or \0>. We have \\\\> and =L+\+\> b\*\>>. In particular, the number > is log-atomic. If \0>, this contradicts the fact that \b>. If \0>, then \log *\>|)>>|)>> implies <\equation*> log b=\\log *\>|)>>|)>. But then > is not a monomial: a contradiction. Assume now that 0>. So =b+n> and \>. But then > is not defined: a contradiction. We conclude that >\L\> *\>>. If \1>, or if =1> and is not tail-atomic, then our claim yields the result. Assume now that =1> and that \\*\> is tail-atomic where \,\\>>, and >=L> E>\>>> is a hyperserial expansion. Then the hyperserial expansion of is >*> E>|)>>>. We next show that \1>>. If \\>*> 2>|)>>>, then\\>, and we conclude with Lemma that \1>>. Assume for contradiction that\\>*> 2>|)>>>. Since > is log-atomic, we must have =0>. By the definition of coding sequences, this implies that =1> and =1>. So \\*exp|)>>, whence =\>, =\>, and =exp|)>>. In particular the number > is atomic, hence tail-atomic. Since \2>>, the claim in the second paragraph of the proof, applied to 1>>, gives >\>>. But then also \>>: a contradiction. We pursue with two auxiliary results that will be used order to construct a infinite path required in the proof of Theorem below. <\lemma> For >, there is a finite path in with >\1>-\> or >\1>-\>>. <\proof> By Lemma, it is enough to find such a path in > a>. Write\\>>. Assume first that =0>, so =1> and =0>. If |)>>> is not tail-atomic, then the hyperserial expansion of |)>>> is |)>>=E|)>>>>> and |)>>>>> is the dominant term of > for some \>>. Then the path with =1> and \r*E|)>>>>> satisfies >=|)>>\1>>. If |)>>> is tail-atomic, then there exist \1>>, \> and \>> such that the hyperserial expansion of |)>>> is |)>>=\>*\>>. Let >*\>> be a term in> with \>>. Then the path with =1> and \r*\>*\>> satisfies >=\\1>-\>. Assume now that \0>. In view of(), we recall that there are an ordinal \\> and a number > with <\equation*> E> a=E>> E>>|)>>>\\|)>. If > is a limit ordinal, then by Lemma, we have a hyperserial expansion \L> E>>|)>>>>. Let \term \>|)>> and set =\> and \\>, so that is a path in >. By Lemma, there is a subpath in > a>, hence also a path in > a>, with -1>=\>. So >=\>|)>\1>>. If > is a successor ordinal, then we may choose =\>*n> for a certain \>. By Lemma, we have a hyperserial expansion \E>>|)>-n>>>. As in the previous case, there is a path in > a> with >=\>, whence >=\>|)>-n\1>-\>. <\corollary> For > and \>, there is a finite path in with \k> and >\k>-\> or >\k>-\>. <\proof> This is immediate if . Assume that the result holds at and pick a corresponding path with >\k>-\> (resp. >\k>-\>). Note that the dominant term > of >-\> (resp. >-\>) lies in *\>*> k+1>|)>>> by Lemma. Moreover > is a term of >>(resp. >>). By the previous lemma, there is a path in > with >\k+1>-\> or >\k+1>-\>, so ,\,P-1|)>,Q|)>\Q> satisfies the conditions. <\theorem> There is a \> such that k>> is nested. <\proof> Assume for contradiction that this is not the case. This means that the set > of indices \> such that we do not have d>=\+\*\>*> d+1>|)>>> is infinite. We write =\i\\|}>> where \d\\>. Fix > and let d\\>. Let d+1>> such that <\equation> \+\*\>*> u|)>>\d>, let \> and let be any finite path with <\equation*> u>=\+\*\>*> u|)>>-n. We claim that we can extend to a path with \>, >\d>-\> and such that > is a bad index in . Indeed, in view of for d>>, the relation () translates into the following three possibilities: <\itemize-dot> There is an \supp \> with \log E> u>. We then have > a\\\log E> u>. By Lemma and the convexity of d+1>>, we deduce that *|)>>*\> lies in the class *log E> d+1>>, so |)>>*\>\> d+1>|)>>>. By Corollary for the admissible sequence starting with ,\|)>> and followed by d+1>>, there is a finite path > in |)>>*\>> with |\|>\d-d\2> and,|\|>>\d>-\>. Taking the logarithm and using Lemma, we obtain a finite path> in |)>>*\>, hence in >, with |\|>\2> and ,|\|>>=u,|\|>>\d>-\>. Write > a|)>>=r*\\\> where \>> and \>>. Then \E> a\supp \>, so the hyperserial expansion of >*\> has one of the following forms <\eqnarray*> >*\>||\\>*> E>|)>>or>>>|>*\>||\\>|)>>>>>> where > E>|)>>> is a hyperserial expansion and > is purely large. In both cases, the path *r*\>*\|)>\R> is a finite path in *\>*> a|)>>> with >=u,|\|>>\d>-\>>. Since > is a term in >>, we may consider the path P\R>. Moreover, since >> is a term in =\>>, the index > is bad for. We have > u\supp \>, but there is an \supp \> with \\>*> u|)>>>. We then have >*> a|)>>\\>*\\\>*> u|)>>>. By Lemma and the convexity of d+1>>, we deduce that |)>>*\> lies in >*> d+1>|)>>>. So >>*|)>>*\|)>>|)>> lies in d+1>>. But then also \>>>*|)>>*\|)>>|)>|)>> lies in d+1>> by Corollary. By Corollary, there is a finite path > in with |\|>\2> and ,|\|>>\d>-\>>. Applying Lemma to this path > in , we obtain is a finite path > in >*|)>>*\|)>>> with ,|\|>>\d>-\>. Since |)>>*\\\>*> d+1>|)>>>, we have \\>*|)>>*\>. So Lemma implies that there is a finite path in |)>>*\>, hence in >, with >\d>-\>. We have \term \\\\term u>>, so P\R> is a path. Write > for the dominant term of *\>*> u|)>>>. The index > is a bad in because >> and > both lie in >>, and >\\>.\ We have > u\supp \> and \\>*> u|)>>>, but =\>\\*\>*> u|)>>|)>>. By the definition of >-truncated numbers, there is a \\> with <\equation*> \>*> u|)>>\> E>>>\\>*> a|)>>. Using the convexity of d+2>>, it follows that > E>>\\>*> d+2>|)>>>. By similar arguments as above (using Corollary and Lemmas and), we deduce that there is a finite path in > with >\d>-\>. As in the previous case P\R> is a path and > is a bad index in. Consider a d-1>> and the path \>>|)>> in . So is a finite path with ,|\|>>\d>>. Thus there exists a path > which extends > with ,|\|>>\d>>, where |\|>> is a bad index in . Repeating this process iteratively for >, we construct a path > that extends > and such that ,|\|>>\d>> and such that |\|>> is a bad index in >. At the limit, this yields an infinite path in that extends each of the paths >. This path has a cofinal set of bad indices, which contradicts Theorem. We conclude that there is a \> such that k>> is nested. <\lemma> Assume that > is nested. Then we have =\+\*\>*>> 1>|]>|)>>>. <\proof> Note that >> 1>|]>=E> \>1>|]>>. The result thus follows from Corollary and the assumption that > is nested. <\lemma> Assume that > is nested. Let \>, > and \> with <\equation> c=\\\*\>*\> for a certain \>> with \E> a> and \\>> a|]>> whenever =0>. If \k>>, then we have <\equation*> |)>\a. <\proof> The proof is similar to the proof of Lemma. We have =\\\*\>*> a|)>>> and we must have \log \> since =\\\*\>*\>\k>>. If follows from the deconstruction lemmas in that \a>. This proves the result in the case when. Now assume that 0>. Setting \\|)>>, let us prove by induction on k>>that <\eqnarray*> >|>|k-p>>>|>|>|,\>>>|>|>|.>>>> For , the last relation yields the desired result. If , then we have \k>> by assumption and we have shown above that \a>>. We have \\>|)>> and >*\>> is amonomial, so () yields =\>|)>\,\>>>. This deals with the case . In addition, we have \0>> because 0> and \k>>. Let us show that <\equation> log c\a. If \0>, then this follows from the facts that \a> and \c>. If =0> and \0>, then /\|)>\\\log /\|)>\a>. If =\=0>, then =E> a> and =\\\>|]>>, so \a>. Assume now that p\k> and that the induction hypothesis holds for all smaller . We have <\equation> c=\|)>=\+\*\>*>>>|)>> Since > is nested, we immediately obtain \\>|)>>, whence \,\>> as above. Since \>> and > is nested, we have \>>. Using(),(), and the decomposition lemmas, we observe that the relation \a> is equivalent to <\equation> E>>>\E> a. We have \a>, so \\>|)>>. Note that <\equation*> E>>|)>>>=\>> a|)>\E> a. So it is enough, in order to derive , to prove that >>>\E>>|)>>>>. Now <\equation*> L> c\\> \>|)> by Lemma, whence >>>\E>>|)>>>> by Lemma. For \>, \>> and >, we have +\*\>*g> a|)>>\i>> by Lemma. We may thus consider the strictly increasing bijection <\equation*> \\\;a\+\*\>*g> a|)>>|)>. We will prove Theorem by proving that the function group \\i\\,g\\>|}>> on > generates the class >, i.e. that we have =>>. We first need the following inequality: <\lemma> Assume that > is nested. Let \> with j> and let \>>. On >, we have \\>> if =1> and \\>>> if =-1>. <\proof> It is enough to prove the result for . Assume that =1>. Let > and set \>|)>>, so that <\eqnarray*> >||+\*\>*> a|)>>>>|>||+\*\>*> a|)>>.>>>> Note that <\eqnarray*> |)>>|>|>|]>.>>>> If =1>, then *\=\/\=1> and > is strictly increasing. So we only need to prove that >|]>\a>, which reduces to proving that >|)>\\>|)>>. Let > be the dominant term of > a>. Our assumption that > is nested gives +\*\>*> a|)>>\i>>, whence \\>|)>>. We deduce that +\*\>*|)>>\\>|)>>. implies that +\*\>*|)>>> is >-truncated.\ <\eqnarray*> >|)>-\>|>|*\>*\>,>>|>|)>-\>|>|*\>*|)>>>>>> and *\=1> implies that <\equation*> \*\>*|)>>-\*\>*\> is a strictly positive term. We deduce that >|)>-\\\>|)>-\>, whence >|)>\\>|)>>. The other cases when =-1> or when =-1>> are proved similarly, using symmetric arguments. \ We are now in a position to prove the following refinement of . <\theorem> If > is nested, then > is a surreal substructure with =>>. <\proof> By Proposition, the class >> is a surreal substructure, so it is enough to prove the equality. We first prove that >\>. Assume for contradiction that there are an >> and a \>, which we choose minimal, such that > cannot be written as =\\\*\> where =\>*>>|)>>> is a hyperserial expansion. Set \\-\>>, |)>>> and \|)>\>>. Our goal is to prove that there is a number > and \>> with <\equation> |>|>|>> a|]>>>|>|>|> a>>|>|>|> a,|\=0|\>> and |r\|\>>.>>>>>> Assume that this is proved and set \\+\*\>*\>>. The first condition and Lemma yield \m>> and the relations \supp \> and >*\>\supp \>. The second and third condition, together with Lemma, imply |)>\a>. The first condition also implies that \>: a contradiction. Proving the existence of and> is therefore sufficient. If \min supp a> or =min supp a> and >, then k> and \\> a>> satisfy(). Assume now that =min supp a> and that >, whence >. If \,\>> then k> and \E>>|)>>>> satisfy(). Assume therefore that \,\>>. This implies that there exist \\> and\*\>> with >>>=L> \.> By the definition of coding sequences, there is a least index k> with \0> or \0>, so <\equation*> E>>>=E+\+\>\\*\>*>>>|)>>|)>\*\>. We have \*\>> and > \\>\*\>>. So by , we must have =\+1>>> for a certain \> and =|)>>*n> for a certain \>>. Note that=L> \-n>. Recall that \a> and > \\>>, so \> \,0|}>>. The case =L> \> cannot occur for otherwise <\equation*> a=-\|\*\>>|)>>=>|\*\>> would not lie in ,\>>. So =0>. Let k+1> and <\equation*> \\> \|\>>|)>>=>|\>>|)>>=\> a>. We have \\>> a|]>> and \E> a>, so and > satisfy(). We deduce that >> is a subclass of >. Conversely, consider > and set \>>. So there are ,i\\> and \\>>\\>>> with ,g>\c\\,g>>. Let max+1,i+1|)>>. By Lemma, there exist ,d\,2|}>> with ,g>\\>>> and ,g>\\>>>, whence >>\c\\>>>. Since > is strictly monotonous, we get -\\b-\>. The numbers *-\|)>> and *-\|)>> are monomials, so -\=b-\>. Therefore>>. In view of Theorem, Lemma, and Proposition, we have the following parametrization of >: <\equation*> |\z\|\>,\> z=,\> \> z\|\,\> \> z,R|}>. We conclude this section with a few remarkable identities for >>. <\lemma> If > is nested, then for \> and >, we have b\a\b>. <\proof> By 4.5> and since the function > is strictly monotonous, it is enough to prove thata,b\,a\b\a\b>. By induction, we may also restrict to the case when . So assume that \b>. Recall that > a\\> b> by Lemma. Since ,b\,\>>, we deduce with Lemma that>>>\E>>>>. It follows using the decomposition lemmas that b>. <\proposition> If > is nested, then we have =1>|)>=\+\*\>*>1>>|)>>>. <\proof> We have \1>|)>> by definition of >. So we only need to prove that 1>|)>\>>. Consider1>>. Since > is nested, the number \+\*\>*> b|)>> is >-admissible, so we need only justify that > b\>\L\> *\>>. Since is >-admissible, we have \\>>. But is 1>>-nested, so \\> for a certain term >. We deduce that >\,\>>, whence > b\>>. Assume for contradiction that >\L\> *\>> and write >=L> \> where \*\>> and \\>. Note that \0>: otherwise > and > would be zero for all 1>, thereby contradicting Definition(). By Corollary, we must have =\+1>> for a certain ordinal> and =\>*n> for a certain \>>. Consequently, > \-n\-n>. If \0>>, then the condition \\>>> implies =b>, which leads to the contradiction that=0\,\>>. If =0>>, then 1>\\*>>, whence >: a contradiction. <\corollary> If > is nested, then for >, we have <\equation*> \> z=\+\*\>*>1>> \ z>>|)>>. <\corollary> If > is nested and \>, then <\equation*> \>=\\\k>>\H>. <\proposition> Assume that > is nested with ,\,\,\|)>=>, assume that \\+1>>> and write \|)>>>. Consider the coding sequence > with ,\,\,\,\|)>=,\,\,\,\|)>> for all \>, with the only exception that <\equation*> \=\-n. If \0>, or =0> and =-1>, then > is nested and we have <\equation*> \>>=L*n>\\>, where > is the class of >>-nested numbers. <\proof> Assume that \0>, or =0> and =-1>. In particular, if is >-admissible, then -\\1>, so -\\supp \>. For ,\>>, it follows that >> is >>-admissible if and only if > b> is >-admissible. Let i>> be the class of i>>admissible numbers, for each \>. We have *n> => by the previous remarks, and > is admissible. For 1>, we have i>=\i>>, so <\equation*> i>=i>\\+\*\>*> i+1>|)>>. Moreover, 1>=1>-n>, so <\eqnarray*> >|>|*n> \L*n> E> 1>=L*n> E> 1>+n|)>=E>> 1>>>|1>>|>|-n+\*\>*> 2>|)>>=\+\*\>*>> 2>|)>>.>>>> So > is nested. We deduce that *n> =>, that is, we have a strictly increasing bijection *n>:\>. It is enough to prove that for > with b>, we have *n> a\L*n> b>. Proceeding by induction on , we may assume without loss of generality that . By, the function >> has the following equation on >>: <\equation*> |\\\>|\>,L> \=> \>>L> \>>,\|}>>>. So it is enough to prove that > b\a>. Note that > b=E>-1>> and >>> where -\,a-\\1>. So -a\1>, whence -1\a>. This concludes the proof. Let > be a number. We say that is if there exists an infinite path in without any bad index for . In that case, yields a coding sequence> which is admissible due to the fact that > with the notations from . By Theorem, we get a smallest \> such that |)>k>> is nested. If , then we say that is . In that case, Theorem ensures that the class > of >-nested numbers forms a surreal substructure, so can uniquely be written as >> for some surreal parameter >. One may wonder whether it could happen that 0>. In other words: do there exist pre-nested numbers that are not nested? For this, let us now describe an example of an admissible sequence >> such that the class >>> of >>-nested numbers contains asmallest element . This number is pre-nested, but cannot be nested by Theorem. Note that our example is \Ptransserial\Q in the sense that it does not involve any hyperexponentials. <\example> Let =,\,0,1,1|)>\>> be a nested sequence with =-1>. Let be the simplest >-nested number. We define a coding sequence >=>,\>,0,1,1|)>\>> by <\eqnarray*> >>|>|>|>>|>|-*\>>>>|>,\>|)>>|>|,\|)>for all |i\0|\>>.>>>>> Note that <\equation*> a=\-\>=\\\*\>, where >> is an infinite monomial, so \>-\>> is >>-nested. In particular, the sequence >> is admissible. Assume for contradiction that there is a >>-nested number with b>. Since >=\>=-1>>, we have \b>. Recall that > and > are purely large, so >\\>=\>>. In particular <\equation*> \>=\-\>>\\-*\>>=\>, which contradicts the assumption that is >>-nested. We deduce that is the minimum of the class >>> of >>-nested numbers. In view of Theorem, the sequence >> cannot be nested. The above examples shows that there exist admissible sequences that are not nested. Let us now construct an admissible sequence >> such that the class >>> of >>-nested numbers is actually empty. <\example> We use the same notations as in . Define >,\>|)>\,1|)>>> and set >,\>|)>\>,\>|)>> for all 0>. We claim that the coding sequence >\>,\>,0,1,1|)>>\>> is admissible. In order to see this, let \*\>>>. Then <\equation*> \>\\*\>=\>\\>*\>\\>\\>*\>>=\. Since >\\*\> is >|)>1>>-admissible (i.e. >>-admissible), we deduce that +\>\\*\>> is >>-admissible, whence >> is admissible. Assume for contradiction that >>> is non-empty, and let \\\>>>. Then > is >>-nested, so \b>, whence \\>: a contradiction. Traditional transseries in can be regarded as infinite expressions that involve , real constants, infinite summation, exponentiation and logarithms. It is convenient to regard such expressions as infinite labeled trees. In this section, we show that surreal numbers can be represented similarly as infinite expressions in > that also involve hyperexponentials and hyperlogarithms. One technical difficulty is that the most straightforward way to do this leads to ambiguities in the case of nested numbers. These ambiguities can be resolved by associating a surreal number to every infinite path in the tree. In view of the results from , this will enable us to regard any surreal number as a unique hyperseries in >. <\remark> In the case of ordinary transseries, our notion of tree expansions below is slightly different from the notion of tree representations that was used in. Nevertheless, both notions coincide modulo straightforward rewritings. Let us consider the monomial =exp> \->+L+1> \|)>> from Example. We may recursively expand > as <\equation*> \=\>>>> \+1>-E*L \>>*> \|)>. In order to formalize the general recursive expansion process, it is more convenient to work with the unsimplified version of this expression <\equation*> \=\\\>>\\>> \|)>+1\1>|)>+\\>\\\\ \|)>>|)>>*> \|)>. Introducing :x\x> as a notation for the \Ppower\Q operator, the above expression may naturally be rewritten as a tree: <\equation*> |gr-frame|>|gr-geometry||gr-grid||gr-grid-old||1>|gr-edit-grid-aspect|||>|gr-edit-grid||gr-edit-grid-old||1>|gr-grid-aspect|||>|gr-grid-aspect-props|||>|gr-text-at-halign|center|gr-transformation||||>|magnify|0.707106780759852||>|>>||>||>>|>|>>|>|>>||>>||>>||>||>||>||>>|>|>>||>>||>||>||>>||>>|>>||>>||>>|>|>>||>>|>|>>||>>|>>||>>||>>||>|>|>>||>||>||>||>||>||>||>>||>|>>||>>|>>||>||>||>>|>|>>||>||>||>|>|>>||>||>||>||>||>||>||>||>||>||>||>>||>>|>|>>||>>||>>||>>||>>||>>|>|>>||>>|>|>>||>>||>||>||>||>||>||>||>||>||>||>>> In the next subsection, we will describe a general procedure to expand surreal monomials and numbers as trees. In what follows, a is a set of nodes > together with a function that associates to each node \N> an >\> and a sequence |]>|)>\\>>\N>>> of ; we write >\|]>\\\\>|}>> for the set of children of >. Moreover, we assume that > contains aspecial element >, called the of , such that for any \N> there exist aunique (called the of > and also denoted by >>) and unique nodes ,\,\>> with =\>, =\>, and \C>> for,h>>. The height > of the tree is the maximum of the heights of all nodes; we set \\> if there exist nodes of arbitrarily large heights. Given a class >, an >-labeled tree> is a tree together with a map :N\\;\\\>>, called the . Our final objective is to express numbers using >-labeled trees, where <\eqnarray*> >|>|>\,>,\,\,\|}>\L>>\E>>.>>>> Instead of computing such expressions in a top-down manner (from the leaves until the root), we will compute them in a bottom-up fashion (from the root until the leaves). For this purpose, it is convenient to introduce a separate formal symbol > for every >, together with the extended signature <\eqnarray*> >>|>|\\c\|}>.>>>> We use > as a placeholder for a tree expression for whose determination is postponed to a later stage. Consider a >>-labeled tree and a map \>. We say that is an of if for each node \N> one of the following statements holds: <\description> >\\>\|}>>, >=0>, and |)>=\>>; >=>>, the family |]>|)>|)>\\>>> is well based and |)>=\\>v|]>|)>>; >=>>, >=2>, and |)>=v|)>*v|)>>; >=\>>, \>, >=1>, and |)>=v|)>>>; >=L>>>, >=1>, and |)>=L>> v|)>>; >=E>>>, >=1>, and |)>=E>> v|)>>; >=>>, >=0>, and |)>=\>. We call |)>> the of via . We say that > is of if there exists an evaluation of with |)>>. <\lemma> <\enumerate-alpha> If has finite height, then there exists at most one evaluation of . Let and > be evaluations of with |)>=v|)>>. Then >. <\proof> This is straightforward, by applying the rules >\U> recursively (from the leaves to the root in the case of (>) and the other way around for (>)). Although evaluations with a given end-value are unique for a fixed tree , different trees may produce the same value. Our next aim is to describe a standard way to expand numbers using trees. Let us first consider the case of a monomial \>. If =1>, then the of > is the >>-labeled tree with =|}>> and >=1>. Otherwise, we may write =\>*> g|)>>> with > or >>. Depending on whether >> or >>, we respectively take <\equation*> T\|gr-frame|>|gr-geometry||gr-grid||gr-grid-old||1>|gr-edit-grid-aspect|||>|gr-edit-grid||gr-edit-grid-old||1>|gr-grid-aspect|||>|gr-grid-aspect-props|||>|gr-text-at-halign|center||>>||>>|>|>>|>|>>|>|>>||>>||>||>||>||>||>>>T\|gr-frame|>|gr-geometry||gr-grid||gr-grid-old||1>|gr-edit-grid-aspect|||>|gr-edit-grid||gr-edit-grid-old||1>|gr-grid-aspect|||>|gr-grid-aspect-props|||>|gr-text-at-halign|center||>>||>>|>|>>|>|>>|>|>>||>||>||>||>||>|>|>>||>>||>>> and call the of >. Let us next consider a general number >> and let \> be the ordinal size of its support. Then we may write \\>c>*\>> for asequence >|)>\\>\>|)>>> and a>decreasing sequence >|)>\\>\>>. For each \\>>, let>> be the standard monomial expansion of >>. Then we define the >>-labeledtree <\equation*> T\|gr-frame|>|gr-geometry||gr-grid||gr-grid-old||1>|gr-edit-grid-aspect|||>|gr-edit-grid||gr-edit-grid-old||1>|gr-grid-aspect|||>|gr-grid-aspect-props|||>|gr-text-at-halign|center|>|>>||>>||>>||>>||>>||>>||>>||>>||>>||>>||>>||>||>||>||>||>||>||>||>||>||>||>>>> and call it the of . Note that the height of is at most , there exists a unique evaluation \> of , and |)>=a>. Now consider two trees and> with respective labelings \ :N\\>> and :N>\\>>>. We say that > if >\N> and there exist evaluations \>> and :N>\>> such that |)>=v|)>> for all \N> and >=\>> whenever >\>>. Now assume that |)>=a> for some evaluation \>. Then we say that is a of if for every \N> with >=>>, the subtree > of with root > refines the standard expansion of |)>>. In particular, a tree expansion of a number > with >\>> always refines the standard expansion of . <\lemma> Any > has a unique tree expansion with labels in >. <\proof> Given \>, we say that an >>-labeled tree is -settled> if >\>> for all nodes \N> of height n>. Let us show how to construct a sequence |)>\>> of >>-labeled tree expansions of such that the following statements hold for each \>: <\description> > is an -settled and of finite height; >|)>=a> for some (necessarily unique) evaluation :N>\>> of >; If 0>, then > refines >; If is a tree expansion of with labels in >, then refines >. We will write :N>\\>> for the labeling of >. We take > such that >=>|}>> and >>=>. Setting >|)>\a>, the conditions>, >, >, and> are naturally satisfied. Assume now that > has been constructed and let us show how to construct >. Let be the subset of >> of nodes > of level with |)>\>>. Given \S>, let >> be the standard expansion of |)>> and let >> be the unique evaluation of >>. We define > to be the tree that is obtained from > when replacing each node \S> by the tree>>. Since each tree >> is of height at most , the height of> is finite. Since > is clearly >-settled, this proves >. We define an evaluation :N>\\>> by setting |)>=v|)>> for any \N>>> and |)>=v>|)>> for any\S> and\N>>> (note that > is well defined since >>>|)>=|)>>=v|)>> for all\S>>). We have >|)>=v>|)>=a>, so > holds for >. By construction, >\N>> and the evaluations > and> coincide on >>; this proves >. Finally, let be a tree expansion of with labels in> and let be the unique evaluation of with |)>=a>. Then refines>, so coincides with > on >>. Let \S>. Since is a tree expansion of , the subtree> of with root > refines >>, whence \N>>>. Moreover, \ |)>=v|)>>, so coincides with >> on >>. Altogether, this shows that refines >. Having completed the construction of our sequence, we next define a >-labeled tree>> and a map >:N>>\> by taking >>=\>N>> and by setting >|)>>\|)>>> and >|)>=v|)>> for any \> and \N>> such that |)>>\>>. By construction, we have >>>|)>=a> and >>refines> for every \>. We claim that >> is a tree expansion of . Indeed, consider a node \N>>> of height with >=>>. Then \N>> and |)>>=>>>, since > is >-settled. Consequently, the subtree of > with root > refines the standard expansion of |)>>. Since>> refines>, it follows that the subtree of >> with root > also refines the standard expansion of >|)>=v|)>>. This completes the proof of our claim. It remains to show that >> is the unique tree expansion of with labels in >. So let be any tree expansion of with labeling :N\\>. For every \>, it follows from > that \N>>. Moreover, since > is -settled, > coincides with both > and >> on those nodes in >> that are of height n>. Consequently, \N>>> and > coincides with >> on>>>. Since every node in > has finite height, we conclude that >>. From now on, we only consider tree expansions with labels in >, as in . Given a class > of nested numbers as in, it can be verified that every element in > has the same tree expansion. We still need a notational way to distinguish numbers with the same expansion. Let > be a pre-nested number. By Theorem, we get a smallest \> such that |)>k>> is nested. Hence \> for the class > of |)>k>>-nested numbers. implies that there exists a unique number with =\>*c|)>>. We call the of and write\c>. By , we note that >=\*\> for all \>. Given an arbitrary infinite path in a number >, there exists a 0> such that k>> has no bad indices for > (modulo a further increase of , we may even assume > to be nested). Let \sign *\*r|)>*\*\*\\>. We call \\*\>> the of , where we note that the value of *\>> does not depend on the choiceof. Let be the tree expansion of a number > and let \> be the evaluation with |)>>. An in is a sequence ,\,\>> of nodes in > with =\> and \C>> for all \>. Such a path induces an infinite path in : let \i\\> be the indices with >>=>>; then we take =v+1>|)>> for each \>. It is easily verified that this induces a one-to-one correspondence between the infinite paths in and the infinite paths in . We call >\\> the of the infinite path =|)>\>> in . Denoting by> the set of all infinite paths in , we thus have a map :I\;\\\>>. We call|)>> the of . We are now in a position to prove the final theorem of this paper. <\render-proof> Proof of <|render-proof> Consider two numbers \> with the same hyperserial description |)>>> and let :N\> be the evaluations of with |)>=a> and |)>=a>>. We need to prove that >. Assume for contradiction that a>. We define an infinite path ,\,\> in with |)>\v|)>> for all by setting \\> and \\>>, where \> is minimal such that |)>\v|)>>. (Note that such a number indeed exists, since otherwise |)>=v|)>> using the rules >\U>.) This infinite path also induces infinite paths and > in and > with =v>|)>> and ,n>=v>|)>> for a certain sequence \i\\> and all \>. Let 0> be such that n>> and n>> have no bad indices for > and ,n>>. The way we chose ,\,\> ensures that the coding sequences associated to the paths n>> and n>> coincide, so they induce the same nested surreal substructure>. It follows that >|)>=a=\>*\>|)>=a,n>=v>|)>>, which contradicts our assumptions. We conclude that and > must be equal. <\bibliography|bib|tm-plain|bibi_save> <\bib-list|38> M.Aschenbrenner, L.vanden DriesJ.vander Hoeven. . 195Annals of Mathematics studies. Princeton University Press, 2017. M.Aschenbrenner, L.vanden DriesJ.vander Hoeven. On numbers, germs, and transseries. , 2, 19\U42. Rio de Janeiro, 2018. M.Aschenbrenner, L.vanden DriesJ.vander Hoeven. Hardy fields, the intermediate value property, and >-freeness. F.Delon, M.Dickmann, D.GondardT.Servi, , 93. 2019. M.Aschenbrenner, L.vanden DriesJ.vander Hoeven. The surreal numbers as auniversal -field. , 21(4):1179\U1199, 2019. V.BagayokoJ.vander Hoeven. Surreal substructures. HAL-02151377 (pre-print), 2019. V.BagayokoJ.vander Hoeven. The hyperserial field of surreal numbers. , HAL, 2021. . V.Bagayoko, J.vander HoevenE.Kaplan. Hyperserial fields. (notes non publiées), 2020. V.Bagayoko, J.vander HoevenV.Mantova. Defining a surreal hyperexponential. , HAL, 2020. . A.Berarducci. Surreal numbers, exponentiation and derivations. , 08 2020. A.BerarducciV.Mantova. Surreal numbers, derivations and transseries. , 20(2):339\U390, 2018. A.BerarducciV.Mantova. Transseries as germs of surreal functions. , 371:3549\U3592, 2019. N.Bourbaki. . Éléments de Mathématiques (Chap. 5). Hermann, 2-nd, 1961. G.Cantor. . Jacques Gabay, 1899. Reprint from les Mémoires de la Société des Sciences physiques et naturelles de Bordeaux. J.H.Conway. . Academic Press, 1976. B.DahnP.Göring. Notes on exponential-logarithmic terms. , 127(1):45\U50, 1987. L.vanden DriesPh.Ehrlich. Fields of surreal numbers and exponentiation. , 167(2):173\U188, 2001. L.vanden Dries, J.vander HoevenE.Kaplan. Logarithmic hyperseries. , 372(7):5199\U5241, 2019. L.vanden Dries, A.MacintyreD.Marker. Logarithmic-exponential series. , 111(1-2):61\U113, 2001. P.duBois-Reymond. Sur la grandeur relative des infinis des fonctions. , 4(1):338\U353, 1870. P.duBois-Reymond. Über asymptotische Werte, infinitäre Approximationen und infinitäre Auflösung von Gleichungen. , 8:363\U414, 1875. P.duBois-Reymond. Über die Paradoxen des Infinitärscalcüls. , 11:149\U167, 1877. J.Écalle. . Hermann, collection: Actualités mathématiques, 1992. J.Ecalle. The natural growth scale. , CARMA, vol 1:93\U223, 2016. U.Felgner. Die Hausdorffsche Theorie der >>-Mengen und ihre Wirkungsgeschichte. Springer-Verlag, gesammelte Werke>, II, 645\U674. Berlin, 2002. G.Fisher. The infinite and infinitesimal quantities of du Bois-Reymond and their reception. , 24:101\U163, 1981. H.Gonshor. . Cambridge Univ. Press, 1986. H.Hahn. Über die nichtarchimedischen Gröÿensysteme. , 116:601\U655, 1907. G.H.Hardy. . Cambridge University Press, 1910. G.H.Hardy. Properties of logarithmico-exponential functions. , s2-10(1):54\U90, 1912. J.vander Hoeven. . , École polytechnique, Palaiseau, France, 1997. J.vander Hoeven. Transséries fortement monotones. Chapter 1 of unpublished CNRS activity report, , 2000. J.vander Hoeven. , 1888. Springer-Verlag, 2006. H.Kneser. Reelle analytische Lösungen der Gleichung |)>=e> und verwandter Funktionalgleichungen. , 187(1/2):56\U67, 1950. D.Lemire. . , Université de Montréal, 1996. D.Lemire. Décompositions successives de la forme normale d'un surréel et généralisation des >-nombres. , 21(2):133\U146, 1997. D.Lemire. Remarques sur les surréels dont la forme normale se décompose indéfiniment. , 23(1):73\U85, 1999. V.MantovaM.Matusinski. Surreal numbers with derivation, Hardy fields and transseries: a survey. , 265\U290, 2017. M.C.Schmeling. . , Université Paris-VII, 2001. <\the-index|idx> > > \>>>-atomic number|> > > > > > > > > > > > > > > > > > > > > > > > > >-simple element|> > > > > >-truncated series|> > > > > <\the-glossary|gly> |]>|]>>|field of well-based series with real coefficients over >|> |support of a series|> |set of terms of a series|> >||> \>>|truncation \\>f>*\> of |> >>|1>>|> g>| and g>|> g>|g-f>|> g>|>*\>|> g>|r\\>,\r*>|> g>|g> and f>|> >>|series \> with 1>|> >>|series \> with 1>|> ,\>>|series \> with 0> and 1>|> >|field of logarithmic hyperseries|> \>>|group of logarithmic hypermonomials of force \>|> \>>|field of logarithmic hyperseries of force \>|> |x> \>>|unique series in > with |x> \>|)>\\>>|> >>>|class of ordinals|> >|simplicity relation|> \\>|ordinal sum of > and >|> \>|ordinal product of > and >|> >>|ordinal exponentiation with base > at >|> \\>|\\>> for each exponent > of >|> \\>|\\>> for each exponent > of >|> \\>|*n\\> for all \>|> \\>|n\\,\\\*n>|> \\>|\\\\>|> >|=\+1> if > is a successor and =\> if > is a limit|> >>|>> for =\>>|> >|composition law :\\>>,\>\>>>|> ,\>>|class of >-truncated series|> >>|>-maximal >-truncated truncation of |> >|class of numbers with g,h\\,g a\b\h a>|> >>>>|class of >-simple elements|> >>|projection >>\>>>>|> >|comparison between sets of strictly increasing bijections|> Y>| and are mutually pointwise cofinal|> |X|\>>|function group generated by |> >|translation a+r>|> >|homothety s*a>|> >|power function a>|> >|function group \r\\|}>>|> >|function group \s\\>|}>>|> >|function group \s\\>|}>>|> >|function group |E H L:n\\,s\\>|\>>|> >>|function group ,L\n\\|}>>|> >|value =P> of the path at 1+>|> >|dominant monomial of >|> >|constant coefficient of >|> >|length of a path >*>>|)>>>|> Q>|concatenation of paths|> >|Berarducci and Mantova's nested truncation relation|> >>>|class of admissible numbers|> >>k>>|class of k>>-admissible numbers|> >>>|class of k>>-nested numbers|> >>>|class of >-nested numbers|> <\initial> <\collection> <\attachments> <\collection> <\associate|bib-bibliography> <\db-entry|+15uLTQSm1iIV3hAO|article|DBR1870> <|db-entry> > <\db-entry|+8lDHqURSvmeWxB|article|DBR1875> <|db-entry> > <\db-entry|+8lDHqURSvmeWxC|article|DBR1877> <|db-entry> > <\db-entry|+8lDHqURSvmeWwP|book|Can1899> <|db-entry> > <\db-entry|+1CQ02y1d169CJ0qP|article|Fish81> <|db-entry> > <\db-entry|+1CQ02y1d169CJ0qQ|incollection|Fel02> <|db-entry> > >>-Mengen und ihre Wirkungsgeschichte> gesammelte Werke> > <\db-entry|+2HPjOjBz2I14ll7z|book|H1910> <|db-entry> > <\db-entry|+2HPjOjBz2I14ll80|article|H1912> <|db-entry> > <\db-entry|+8lDHqURSvmeWvv|book|Bour61> <|db-entry> > <\db-entry|+9izXaIC09Kv5s5|book|Con76> <|db-entry> > <\db-entry|+8lDHqURSvmeWz8|book|Gon86> <|db-entry> > <\db-entry|+BacBZ9audkKRMn|article|EvdD01> <|db-entry> Ph. > <\db-entry|+BacBZ9audkKRMt|article|MM17> <|db-entry> M. > <\db-entry|+20um2SMq3lIEL5R|unpublished|Ber20> <|db-entry> > > <\db-entry|+20um2SMq3lIEL5k|article|DG87> <|db-entry> P. > <\db-entry|+8lDHqURSvmeWxr|book|Ec92> <|db-entry> > <\db-entry|+29WgnAxi2VXmdsVM|article|Ec16> <|db-entry> > <\db-entry|+8lDHqURSvmeX7z|phdthesis|vdH:phd> <|db-entry> > <\db-entry|+QfXtZPU0IGdwf6|article|vdDMM01> <|db-entry> A. D. > <\db-entry|+8lDHqURSvmeX8c|book|vdH:ln> <|db-entry> > <\db-entry|+2KXhjR2JqOY31OS|incollection|vdH:hivp> <|db-entry> L. van den J. van der > >-freeness> M. D. T. > <\db-entry|+AZQhoxVyzFxH5u|article|BM19> <|db-entry> V. > <\db-entry|+1CQ02y1d169CJ0pN|inproceedings|vdH:icm> <|db-entry> L. van den J. van der > <\db-entry|+8lDHqURSvmeX0u|article|Kne50> <|db-entry> > |)>=e> und verwandter Funktionalgleichungen> <\db-entry|+1Hcl3U922Lc9q61C|techreport|vdH:surhypexp> <|db-entry> J. van der V. > > <\db-entry|+9izXaIC09Kv5vK|phdthesis|Schm01> <|db-entry> > <\db-entry|+1Hcl3U922Lc9q61L|techreport|vdH:hypno> <|db-entry> J. van der > > <\db-entry|+YXseyEg1Ca84l5v|article|vdH:loghyp> <|db-entry> J. van der E. > <\db-entry|+20um2SMq3lIEL5V|unpublished|BvdHK:hyp> <|db-entry> J. van der E. > <\db-entry|+29WgnAxi2VXmdsVe|unpublished|vdH:gentr> <|db-entry> > > <\db-entry|+20um2SMq3lIEL5T|unpublished|BvdH19> <|db-entry> J. van der > <\db-entry|+dRhQtvw1aqJlV7d|phdthesis|Lemire:phd> <|db-entry> > <\db-entry|+dRhQtvw1aqJlV7e|article|Lemire97> <|db-entry> > >-nombres> <\db-entry|+dRhQtvw1aqJlV7f|article|Lemire99> <|db-entry> > <\db-entry|+K2BzYNZCdNkuzC|article|BM18> <|db-entry> V. > <\db-entry|+2KXhjR2JqOY31Nl|article|vdH:bm> <|db-entry> L. van den J. van der > universal -field> <\db-entry|+TlUcUrl9iqHNw5|book|vdH:mt> <|db-entry> L. van den J. van der > <\db-entry|+BacBZ9audkKRMp|article|Hahn1907> <|db-entry> > <\references> <\collection> > > > > > > > > |\>|7>> > > > > > |P\Q>|22>> > > |\>|24>> > > > > > > > > |||font-series||font-shape||Ad>>>>|34>> |||font-series||font-shape||Ad>>>k>>|34>> > > |f\>>|7>> |||font-series||font-shape||Ne>>>>|35>> |||font-series||font-shape||Ne>>>>|35>> > > > > > > > > |f>>|7>> > |h=f\g>|7>> > |f\g>|7>> |f\g>|8>> |f\g>|8>> |f\g>|8>> > |\>>|8>> |\>>|8>> |\,\>>|8>> > > > > > > |\>|8>> > > |\\>>|8>> |\\>>|9>> |g|x> \>>|9>> > > |||font-series||font-shape||On>>>>|9>> |\>|9>> |\\\>|10>> |a\\>|10>> > |\>>|10>> > > > |\\\>|10>> |\\\>|10>> |\\\>|10>> |\\\>|10>> |\\\>|10>> |\>|10>> > |\>>|10>> > |\>|10>> > > > > > > |L\>>>-atomic number|11>> |\|]>|]>>|7>> > |\>-truncated series|11>> |\,\>>|12>> |\>>|12>> > > > > > > |supp f>|7>> > > > > > |\>|14>> |\>-simple element|14>> |||font-series||font-shape||Smp>>>>>|14>> |\>>|14>> |\>|14>> > |X\Y>|14>> ||X|\>>|14>> > |T>|15>> |H>|15>> |P>|15>> |\>|15>> |\>|15>> |\>|15>> |\>|15>> |term f>|7>> |\>>|15>> > > > > |\>|21>> |\>|21>> |r>|21>> > |>|21>> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > |||font-series||font-shape||C1>>>>|10>> |||font-series||font-shape||C2>>>>|10>> |||font-series||font-shape||C3>>>>|10>> |||font-series||font-shape||C4>>>>|10>> |||font-series||font-shape||C5>>>>|10>> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> DBR1870 DBR1875 DBR1877 Can1899 Fish81 Fel02 H1910 H1912 Bour61 Con76 Gon86 EvdD01 MM17 Ber20 DG87 Ec92 Ec16 Ec92 vdH:phd vdDMM01 vdH:ln vdH:hivp BM19 vdH:ln vdH:icm Kne50 vdH:surhypexp Schm01 vdH:hypno Ec92 Ec16 Schm01 vdH:loghyp BvdHK:hyp BvdHK:hyp vdH:hypno vdH:phd vdH:gentr Schm01 BvdH19 Con76 Lemire:phd Lemire97 Lemire99 BvdH19 vdH:ln BvdH19 vdH:loghyp BvdHK:hyp vdH:surhypexp vdH:hypno vdH:ln BM18 vdH:bm vdH:mt vdH:icm BvdH19 vdH:loghyp BvdHK:hyp vdH:surhypexp vdH:hypno vdH:loghyp Hahn1907 BvdHK:hyp vdH:hypno vdH:hypno Schm01 BM18 vdH:hypno BvdH19 BvdH19 Hahn1907 vdH:loghyp vdH:loghyp vdH:loghyp vdH:loghyp vdH:loghyp vdH:loghyp Gon86 Con76 BvdH19 Con76 vdH:hypno vdH:hypno vdH:hypno BvdHK:hyp BvdHK:hyp vdH:hypno BvdHK:hyp BvdHK:hyp BvdH19 BvdH19 BvdH19 Gon86 vdH:hypno BvdH19 BvdH19 vdH:hypno vdH:hypno vdH:hypno vdH:phd Schm01 BM18 Gon86 Gon86 BvdH19 BM18 BM18 BM18 BM18 BvdHK:hyp BvdH19 BvdH19 BvdH19 BvdH19 BvdH19 BvdH19 BvdH19 vdH:hypno vdH:phd Schm01 <\associate|gly> |\|]>|]>>|field of well-based series with real coefficients over |\>|> |supp f>|support of a series|> |term f>|set of terms of a series|> |\>||max supp f>|> |f\>>|truncation |\\>f>*\> of |f>|> |f>>||f1>>|> |h=f\g>||h=f+g> and |supp f\g>|> |f\g>||supp f\g-f>|> |f\g>||\>*\>|> |f\g>||\r\\>,\r*>|> |f\g>||f\g> and |g\f>|> |\>>|series |f\\> with |supp f\1>|> |\>>|series |f\\> with |f\1>|> |\,\>>|series |f\\> with |f\0> and |f\1>|> |\>|field of logarithmic hyperseries|> |\\>>|group of logarithmic hypermonomials of force |\\>|> |\\>>|field of logarithmic hyperseries of force |\\>|> |g|x> \>>|unique series in |\> with |g=|x> \>|)>\\>>|> |||font-series||font-shape||On>>>>|class of ordinals|> |\>|simplicity relation|> |\\\>|ordinal sum of |\> and |\>|> |a\\>|ordinal product of |\> and |\>|> |\>>|ordinal exponentiation with base |\> at |\>|> |\\\>||\\\>> for each exponent |\> of |\>|> |\\\>||\\\>> for each exponent |\> of |\>|> |\\\>||\*n\\> for all |n\\>|> |\\\>||\n\\,\\\*n>|> |\\\>||\\\\\>|> |\>||\=\+1> if |\> is a successor and |\=\> if |\> is a limit|> |\>>||\>> for |\=\>>|> |\>|composition law |\:\\||font-series||font-shape||No>>>,\>\||font-series||font-shape||No>>>>|> |\,\>>|class of |\>-truncated series|> |\>>||\>-maximal |\>-truncated truncation of |s>|> |\>|class of numbers |b> with |\g,h\\,g a\b\h a>|> |||font-series||font-shape||Smp>>>>>|class of |\>-simple elements|> |\>>|projection |||font-series||font-shape||S>>>\||font-series||font-shape||Smp>>>>>|> |\>|comparison between sets of strictly increasing bijections|> |X\Y>||X> and |Y> are mutually pointwise cofinal|> ||X|\>>|function group generated by |X>|> |T>|translation |a\a+r>|> |H>|homothety |a\s*a>|> |P>|power function |a\a>|> |\>|function group |\r\\|}>>|> |\>|function group |\s\\>|}>>|> |\>|function group |\s\\>|}>>|> |\>|function group ||E H L:n\\,s\\>|\>>|> |\>>|function group |,L\n\\|}>>|> |\>|value |\=P> of the path |P> at |i\1+>|> |\>|dominant monomial of |\>|> |r>|constant coefficient of |\>|> |>|length of a path |P\>*||font-series||font-shape||Mo>>>|)>>>|> |P\Q>|concatenation of paths|> |\>|Berarducci and Mantova's nested truncation relation|> |||font-series||font-shape||Ad>>>>|class of admissible numbers|> |||font-series||font-shape||Ad>>>k>>|class of |\k>>-admissible numbers|> |||font-series||font-shape||Ne>>>>|class of |\k>>-nested numbers|> |||font-series||font-shape||Ne>>>>|class of |\>-nested numbers|> <\associate|idx> |> |> |> |> |> |> |> |> |> |> |> |> |> |> |> |L\>>>-atomic number>|> |\>-truncated series>|> |> |> |> |> |> |> |> |\>-simple element>|> |> |> |> |> |> |> |> |> |> |> |> |> <\associate|toc> |math-font-series||font-shape||1.Introduction> |.>>>>|> |1.1.Toward a unification of infinities |.>>>>|> > |1.2.Outline of our results and contributions |.>>>>|> > |math-font-series||font-shape||2.Ordered fields of well-based series> |.>>>>|> |2.1.Well-based series |.>>>>|> > |2.2.Well-based families |.>>>>|> > |2.3.Logarithmic hyperseries |.>>>>|> > |math-font-series||font-shape||3.Surreal numbers as a hyperserial field> |.>>>>|> |3.1.Surreal numbers |.>>>>|> > |3.2.Hyperserial structure on |||font-series||font-shape||No>>>> |.>>>>|> > |3.3.Hyperlogarithms |.>>>>|> > |3.4.Atomicity |.>>>>|> > |3.5.Hyperexponentiation |.>>>>|> > |math-font-series||font-shape||4.Surreal substructures> |.>>>>|> |4.1.Cuts |.>>>>|> > |4.2.Cut equations |.>>>>|> > |4.3.Function groups |.>>>>|> > |4.4.Remarkable function groups |.>>>>|> > |math-font-series||font-shape||5.Well-nestedness> |.>>>>|> |5.1.Hyperserial expansions |.>>>>|> > |5.2.Paths and subpaths |.>>>>|> > |5.3.Deconstruction lemmas |.>>>>|> > |5.4.Nested truncation |.>>>>|> > |5.5.Well-nestedness |.>>>>|> > |math-font-series||font-shape||6.Surreal substructures of nested numbers> |.>>>>|> |6.1.Coding sequences |.>>>>|> > |6.2.Admissible sequences |.>>>>|> > |6.3.Nested sequences |.>>>>|> > |6.4.Pre-nested and nested numbers |.>>>>|> > |math-font-series||font-shape||7.Numbers as hyperseries> |.>>>>|> |7.1.Introductory example |.>>>>|> > |7.2.Tree expansions |.>>>>|> > |7.3.Hyperserial descriptions |.>>>>|> > |math-font-series||font-shape||Bibliography> |.>>>>|> |math-font-series||font-shape||Index> |.>>>>|> |math-font-series||font-shape||Glossary> |.>>>>|>