INITIAL EMBEDDINGS IN THE SURREAL NUMBER TREE

A Thesis Presented to

The Honors Tutorial College

Ohio University

In Partial Fulfillment of

the Requirements for Graduation

from the Honors Tutorial College

with the degree of

Bachelor of Science in Mathematics

by Elliot Kaplan

April 2015 Contents

1 Introduction 1

2 Preliminaries 3

3 The Structure of the Surreal Numbers 11

4 Leaders and Conway Names 24

5 Initial Groups 33

6 Initial Integral Domains 43

7 Open Problems and Closing Remarks 47 1 Introduction

The infinitely large and the infinitely small have both been important topics throughout the history of mathematics. In the creation of modern calculus, both

Isaac Newton and Gottfried Leibniz relied heavily on the use of infinitesimals.

However, in 1887, Georg Cantor attempted to prove that the idea of infinitesimal numbers is a self-contradictory concept. This proof was valid, but its scope was sig- nificantly narrower than Cantor had thought, as the involved assumptions limited the proof’s conclusion to only one conception of number. Unfortunately, this false claim that infinitesimal numbers are self-contradictory was disseminated through

Bertrand Russell’s The Principles of Mathematics and, consequently, was widely believed throughout the first half of the 20th century [6].Of course, the idea of in-

finitesimals was never entirely abandoned, but they were effectively banished from calculus until Abraham Robinson’s 1960s development of non-standard analysis, a system of mathematics which incorporated infinite and infinitesimal numbers in a mathematically rigorous fashion.

One non-Archimedean number system is the system of surreal numbers, a sys- tem first constructed by John Conway in the 1970s [2]. The surreal numbers are of particular interest because they form (in a sense that can be made precise) the largest possible real-closed ordered field in von Neumann-Bernays-G¨odelset the- ory, one of the most widely used set theories in mathematics [4]. Not only is the system of surreal numbers the largest real-closed ordered field, it contains all other real-closed ordered fields as initial subfields [5].

The surreal numbers, No, form both an ordered group and an ordered field

1 Figure 1: The initial stages of the surreal number tree.

with a full lexicographically ordered binary tree structure where x

is simpler than y and in which sums and products are defined to be the simplest

elements consistent with No’s ordered group and ordered field structure. In this

thesis, I consider a class of s-hierarchical ordered structures, structures first intro-

duced in [5] which generalize No’s simplicity structure and which are all isomorphic

to initial substructures of No. Included in this class are some familiar structures,

such as the field of real numbers and the proper class of ordinals. In the following

sections, I will present theorems stating precisely which ordered groups and or-

dered integral domains are isomorphic to initial subdomains and initial subgroups

of this surreal number tree, revealing them to be s-hierarchical ordered structures.

This extends the theorem on initial subfields published by Philip Ehrlich in [5]. I

2 will also present some additional results regarding initial subdomains of No.

I will begin by introducing some mathematical preliminaries in Section 2, such as sets, functions and the ordinal numbers. I will then introduce the surreal num- bers themselves and some algebraic structures in Section 3. It is in this section that I will introduce the class of s-hierarchical ordered structures. I will discuss

Conway Names and how they provide us with a view of the surreal numbers as a vector space over the real numbers in Section 4. I will present the main theo- rems and their proofs in Sections 5 and 6 and I will end with some open questions and closing remarks in Section 7. Throughout, I will use notation and definitions consistent with those used in [5].

Much of the groundwork for this thesis was established in [5]. The theorems in this thesis were formulated jointly with Philip Ehrlich and many of the results were proved with his aid. Portions of this thesis were presented at the Annual

North American meeting of the Association for Symbolic Logic on March 25, 2015 at the University of Illinois at Urbana-Champaign. In addition to Dr. Ehrlich, I would like to thank Dr. Todd Eisworth for all of his help over the last four years.

2 Preliminaries

2.1 Sets and proper classes

A set is, naively, a collection of objects, a conception that has, since its origin, been made rigorous through various axiomatizations. In this thesis, the underly- ing set theory will be von Neumann-Bernays-G¨odelset theory (NBG). This is a conservative extension of Zermelo-Fraenkel set theory with choice (ZFC), meaning

3 that a statement in the language of ZFC is provable in ZFC if and only if it is

provable in NBG. I will not go through most of the axioms of NBG, but I will

discuss the notion of a “proper class,” a type of entity which is formalized in NBG

and only presented informally in ZFC.

Originally, set theory was a naive theory, meaning that it was not axiomatized using formal logic but, instead, was defined using natural language. In naive set theory, any definable collection of objects can be considered as a set. For example, the collection of all sets could be considered a set. However, in 1901, Bertrand

Russell showed that this naive set theory was self-contradictory as, if any definable collection of objects is a set, then the collection of all sets that are not members of themselves is a set. This set, the Russell set, can be formally written as {x : x∈ / x}.

The problem arises when we ask: is the Russell set a member of the Russell set?

If it is, then it is a member of itself, so it cannot be in the Russell set. However,

if it is not, then it is not a member of itself, so it is in the Russell set. There are

various axiomatizations of set theory which avoid this paradox, but the method

used in NBG set theory is to say that some collections of objects are simply “too

big” to be sets. One such object is the collection of all sets. These objects are

referred to as proper classes in NBG. All proper classes can be put in a one-to-one

correspondence with the proper class of all sets. Two important proper classes in

the context of this thesis are the collection of all ordinals, On, and the proper class

of surreal numbers, No. Throughout this thesis, the term class will mean a set or

a proper class and the term set will be used only when the class in question either

is or must be a set.

The notation x ∈ X means that the object x is an element of the class X.

4 Classes are uniquely defined by their members, meaning that if the set A contains only the member x, written A = {x}, and B = {x} as well, then A = B.A consequence of this is that the set {x, x} is equal to the set {x} (both sets contain exactly the same elements). Another consequence is that there is at most one set which contains no elements. The existence of such a set, called the empty set, is implied by other axioms of NBG. The empty set is referred to by the symbol ∅.A class Y will be said to be a subclass of a class X (written Y ⊆ X) if and only if for every y ∈ Y , y is also in X. Note that by this definition, X ⊆ X. A class Y is a proper subclass of a class X (Y ( X) if Y is a subclass of X which is not equal to X. This means that there is an element of X which is not in Y .

An indexed collection of classes is written as {Xγ : γ ∈ Γ}. This means that for every element γ of the indexing class Γ, there is a corresponding class Xγ in the collection. If this collection both a set and well-ordered, we may write this collection as {Xα : α < β}, meaning for every ordinal α less than some ordinal

β, Xα is in the collection. If this collection is a well-ordered proper class, we may write the collection as {Xα : α ∈ On}, meaning for every ordinal α, Xα is in the collection. The definition of a well-ordering and a construction of the ordinals will S be given later. Given a family of classes, we can construct its union, γ∈Γ Xγ and T its intersection γ∈Γ Xγ. The union is the class of all elements found in any class in the collection and the intersection is the class of elements found in every class in the collection. The union and intersection of two classes, A and B, are written

A ∪ B and A ∩ B, respectively.

An equivalence relation ∼ on a class A is a two-place relation where, given any a, b and c ∈ A:

5 1. a ∼ a (reflexivity).

2. If a ∼ b then b ∼ a (symmetry).

3. If a ∼ b and b ∼ c then a ∼ c (transitivity).

Given an equivalence relation ∼ on A, we can define the equivalence class of an

element a ∈ A, written [a], as the class {b ∈ A : a ∼ b}. Given any a, b ∈ A,

either [a] = [b] or [a] ∩ [b] = ∅, that is, either the equivalence classes of a and b are identical or disjoint. This allows us to break the class A into the disjoint union of equivalence classes under an equivalence relation, which will be useful when the

Conway names are defined in Section 4.

Before introducing functions, I will define a few basic sets which we will be working with. The set of natural numbers is the set {0, 1, 2,...} of counting num- bers. The set of integers is the set {0, 1, −1, 2, −2,...} containing all of the natural numbers as well as their additive inverses. The set of dyadic rationals is the set of

n reduced fractions of the form 2m where n is an integer and m is a natural number.

5 4 1 For example, 16 is a dyadic rational, as 16 = 2 . 3 is not, as 3 is not a power

15 5 of two. 48 is not a reduced fraction, but it reduces to 16 , a dyadic rational. The set of real numbers contains all of the previous sets. It can be axiomatized as an

ordered field for which all of its subsets with an upper bound have a least upper

bound. Of course, these sets can be constructed in many different ways. In fact,

these sets all have definitions in the theory of surreal numbers. For example, the

set of No’s integers is the set of all surreal numbers of finite tree-rank for which

either the left or right set of predecessors is empty (this definition will be more

meaningful after the definitions for tree-rank and predecessors is given). Also of

6 note is that all of these sets are initial subtrees of the surreal numbers. These sets

will be defined within the surreal numbers at the end of Section 3.

2.2 Functions

A function, often called a mapping, is a relation between a class of inputs and a

class of possible outputs. Let f be a function mapping A to B (written f : A → B).

The domain of a f, written D(f), is the class of all inputs for f and the range of the f, written R(f) is the class of all outputs. The domain is equal to the class being mapped from (D(f) = A) and the range is a subclass of the class being mapped to (R(f) ⊆ B). For an input a ∈ D(f), the corresponding output f(a) is an element of R(f). A function is called surjective (or onto) if R(f) = B.A function is called injective (or one-to-one) if for any x 6= y ∈ A, f(x) 6= f(y). A function is bijective if it is both injective and surjective.

An isomorphism is a bijective mapping which preserves some sort of structure between the domain and range. For example, if A and B are fields, then f : A → B is an isomorphism of fields if it is a bijection and for any x, y ∈ A:

1. f(x + y) = f(x) + f(y).

2. f(x · y) = f(x) · f(y).

An embedding of A into B is an isomorphism between A and a subclass of B.

2.3 Ordered classes

An ordered class, hX, ordered pair of elements x, y ∈ X is either true

7 or false. All orders discussed in this section will be strict orders, meaning that

for any x ∈ X, x 6< x (irreflexivity). A non-strict order, usually denoted by ≤, is

defined for x, y ∈ X as follows: x ≤ y if and only if x < y or x = y. Both strict and non-strict orders will be used throughout this thesis, though only an introduction to strict orders should be necessary.

The most fundamental type of order is a partial order. If < is a partial order,

then for any x, y and z ∈ X:

1. x 6< x (irreflexivity).

2. if x < y and y < z, then x < z (transitivity).

In a partially ordered class, some elements may be incomparable (i.e. x 6= y, x 6< y

and y 6< x). If all elements are comparable, then the order is a total order. More

precisely, a total order is a partial order which also satisfies trichotomy, that is, for

any x, y ∈ X, either y < x, x < y or x = y. Note that irreflexivity and transitivity

imply that only one of these can hold.

Finally, we have classes which are well-founded. These are partially ordered

classes which have the following property: every nonempty subclass of the class

has a least element with respect to the partial order (i.e. hX,

if for all Y ⊂ X there is a y ∈ Y such that for all x ∈ Y either x = y or y < x).

For example, the set of all natural numbers is well-founded by its usual order but

the set of all integers is not, as the subset of all negative integers does not have a

least element. If a well-founded class is totally ordered (like the natural numbers),

then it is a well-ordered class. In NBG set theory, all classes can be well-ordered

(a statement which is equivalent to the axiom of global choice).

8 If hA,

phism if f is a bijection and for any x, y ∈ A, x

All well-ordered sets are order isomorphic to a unique ordinal number and all well-ordered proper classes are order isomorphic to the class of ordinals.

2.4 Ordinals and induction

The proper class of ordinals, On, is an extension of the natural numbers which are used to denote the order type of a well-ordered class. Every natural number n is an ordinal number corresponding to a well-ordering of a finite set of n elements. The ordinal corresponding to the natural numbers themselves is ω. In 1923, John von

Neumann provided a definition of the ordinals which is widely used in set theory.

Von Neumann’s ordinals have the following properties:

Zero: The ordinal 0 is defined to be the empty set, ∅.

Successor Ordinals: Given any ordinal α, the ordinal α + 1 is defined to be the

set α ∪ {α}, that is, the set containing all elements of α and α itself.

Limit ordinals: If the ordinals β have been defined for all β < α, then the ordinal

α = {β : β < α}.

Von Neumann’s ordinals are well-ordered by inclusion, meaning that hOn, ∈i is

a well-ordered proper class by von Neumann’s construction. In Figure 1, we can

see that the outer right branch of the surreal number tree corresponds with the

class of ordinals. It should, however, be noted that the elements in this branch do

not have the proterties above. Within the ordered field structure of the surreals,

these numbers do not follow the rules of Cantorian ordinal arithmetic. However,

9 the operations of ordinal arithmetic can be defined for these numbers. Instead,

they are constructed like any other surreal number and adhere to the same field

operations as the rest of No. The class of No’s ordinals, ordered by simplicity is, however, order isomorphic to von Neumann’s ordinals, ordered by inclusion. The simplicity ordering and field operations will be further discussed in Section 3.

Transfinite mathematical induction is a method of mathematical proof which can be used to show that a statement holds for all elements in a well-ordered class.

First, suppose that A be a well-ordered set (resp. proper class). Let f be the order isomorphism mapping A onto some unique ordinal β (resp. onto On). Let us index the elements of A by their corresponding ordinals so that for α < β (resp.

α ∈ On), aα is the unique element of A which is mapped to α by f. We can prove

that a statement P is true for each element in A by proving two different cases:

The initial case: We must prove that P (a0) is true.

The induction step: If α is an arbitrary non-zero ordinal where α < β (resp.

α ∈ On), we must prove that if P (aλ) is true for all λ < α, then P (aα) is

true as well.

Generally, though not always, the second step is split into two cases: one for limit ordinals and one for successor ordinals.

Induction can also be done on a well-founded class. In this case, we show that the least element of a class X satisfies a given property P and then show that for any other element x in the class, if P (y) holds for all y < x then P (x) holds.

Transfinite induction is a special case of this. In the theory of surreal numbers, most proofs by induction are done by induction on a well-founded class.

10 3 The Structure of the Surreal Numbers

3.1 Conway’s construction of the surreal numbers

In the first few pages of On Numbers and Games, Conway provides us with a con-

struction of the surreal numbers. In the following few subsections, it will be shown

that this construction provides us with a full lexicographically ordered binary tree

and an s-hierarchical ordered field, but until these structures are rigorously defined,

the focus will be on the simple, recursive construction of the surreal numbers. This

subsection is essentially a summary of the first chapter of [2].

Any surreal number, Conway says, is simply an entity of the form {L | R} where

L and R are sets of surreal numbers and every member of L is strictly less than every member of R (sometimes written L < R) [2]. These are actually extracted from Conway’s games, which are also entities of the form {L | R} where L and R

are sets of games. Notice that the condition that L < R and the condition that

L and R consist of numbers have been dropped. The surreal numbers, then, are

just special cases of these games, and they can be extracted inductively. Conway

defines a partial order on the games, and when this order is restricted to the surreal

numbers it can be shown to be a total order. Though Conway devotes the second

half of his book to games, this thesis will focus only on the numbers.

At the beginning of Conway’s construction, there are no surreal numbers to

work with. Since it is stipulated that L and R be sets of surreal numbers, they

must both be empty. It is vacuously true that every element in L is strictly less

than every element in R, as neither set contains any elements, so { | } is a well-

defined, legitimate surreal number. Conway refers to this number as 0 without

11 immediate proof that it behaves like 0 in the algebraic sense (i.e., that for any

surreal number x, 0 + x = x and 0 · x = 0). Once the operations of addition and multiplication are defined for surreal numbers, it is easy to prove that 0 has these properties. Of course, the surreal number system can be built without naming any element, but naming the numbers as they are constructed allows us to keep the notation understandable.

Once 0 has been constructed, we have a surreal number to work with. In constructing the next surreal numbers, we have two possibilities: either 0 ∈ L and

R is empty or 0 ∈ R and L is empty. If we were to try to include 0 in both L and

R, then {L | R} would not be a surreal number, as the element 0 ∈ L is not strictly

less than the element 0 ∈ R. If we include 0 in neither L nor R, then both sets are

empty and we will just construct 0 again. We will name these two new numbers

1 = {0 | } and −1 = { | 0}. Once again, it can be shown that these numbers act

like the multiplicative identity and its inverse once Conway’s algebraic operations

are defined.

1 1 Now we can construct the surreal numbers 2 = {0, 1 | }, 2 = {0 | 1}, − 2 = {−1 | 0} and −2 = { | − 1, 0}. Again, once Conway’s algebraic operations are

defined, it is easy to show that these surreal numbers have all of the properties that

1 1 we would normally associate with the numbers 2, 2 , − 2 and −2. It is also worth considering the surreal number {−1 | 1}, which is a legitimate surreal number as every member of the left set is less than every member of the right set. This number is actually equal to 0, which illustrates an important point: every surreal number has multiple representations of the form {L | R}. However, for any sets

L and R of surreal numbers for which L < R, there is one and only one surreal

12 number {L | R}. Conway proves that the surreal number {L | R} is the earlies

number created that lies between L and R. Ehrlich defines this number to be the

simplest surreal number between L and R but in order to discuss simplicity, we must first discuss trees.

3.2 Lexicographically ordered binary trees

A tree hA,

0 0 subclass A of A is an initial subtree of A if for each x ∈ A , prA0 (x) = prA(x). As

prA(x) is a well-ordered class, it is order isomorphic to some ordinal. This ordinal,

ρA(x), is called the tree-rank of x and the αth level of A, LevA(α), is the class

{x ∈ A : ρA(x) = α}. The class LevA(0) is the class of roots of A and the height

of A is either the least ordinal α such that LevA(α) = ∅ or On if LevA(α) 6= ∅

for all α ∈ On.A chain in A is a subclass of A totally ordered by

element y ∈ A, y is an immediate successor of an element x ∈ A if x

ρA(y) = ρA(x) + 1 and an immediate successor of a chain (xα)α<β in A if xα

for all α < β and ρA(y) = min{σ ∈ On : σ > ρA(xα) for all α < β}, that is, the

tree-rank of y is the least ordinal greater than the tree-rank of all xα for α < β.

S β Let B = β

plusses and minuses of any ordinal length. For x, y ∈ B let x

is a proper initial subsequence of y, that is, the sequence of plusses and minuses

denoting x is an initial subsequence of the sequence denoting y. Initial subtrees of

the tree hB,

13 Definition 1. A tree is binary if and only if it is isomorphic to a canonical binary tree.

Equivalently, a tree is binary if and only if each element has at most two immediate successors and each chain of limit length (including the empty chain) has at most one immediate successor [3]. If each element has exactly two immediate successors and each chain of limit length has exactly one immediate successor, then

this binary tree will be said to be full.

If the universe of a tree hA,

said to be an ordered tree. Canonical binary trees can be ordered lexicographically

as follows: consider two members x = (xα)α<µ and y = (yα)α<σ of a canonical

binary tree where each xα, yα ∈ {−, +}. We say that x

some β ≤ min{µ, σ}, where xβ < yβ (where − < undefined < +) but for all

α < β, xα = yα (i.e., β is the least ordinal at which x and y differ). This order is

called a lexicographical order as it generalizes the alphabetical ordering of words.

Definition 2. An ordered binary tree A is lexicographically ordered if there is

an order and tree-rank preserving isomorphism between A and a lexicographically

ordered canonical binary tree.

Equivalently, an ordered binary tree A is lexicographically ordered if for each

x, y ∈ A where x < y, x is incomparable with y if and only if they have a common

predecessor z such that x < z < y [5]. Consider a member x of an ordered tree

hA, <,

of A which are both less than and simpler than x. Likewise, Rs(x) = {y ∈ A : x <

y and y

14 tree and consider two subsets of A, L and R (note that these must be sets, not proper classes), where L < R (every member of L is less than every member of

R). The simplest member of the class {y ∈ A : L < {y} < R} will be denoted by

{L | R}A (assuming that the class of elements between L and R is nonempty). By

Theorem 1 in [5], every nonempty convex subclass of a lexicographically ordered binary tree contains a simplest member. A subclass I of A is convex if for any x, x0 ∈ I where x < x0, all members of A between x and x0 are members of I.

Clearly, {y ∈ A : L < {y} < R} is a convex subclass, so {L | R}A is a well- defined element. If L and R are sets of surreal numbers and A is a class of surreal

numbers, then, as mentioned in the previous subsection, {L | R}A is a surreal number if {y ∈ A : L < {y} < R} is nonempty. Just as every surreal number has

multiple representations, for any lexicographically ordered binary tree hA, <,

and for any x ∈ A, there may be many pairs of subsets L and R of A such that

A A x = {L | R} . The representation {Ls(x) | Rs(x)} is the canonical representation

for x, although many representations will be used.

If x = {L | R}A, then by xL we mean a typical member of L and likewise for

R x . If L and R are not made clear, they will be assumed to be Ls(x) and Rs(x).

We may sometimes write x as {xL | xR}A. The superscript denoting the universe

may be dropped if the universe is clear from context.

The following two theorems, Theorems 2 and 3 from [5] provide us with useful

rules for representing an element x with a non-canonical representation, for proving

that one element is simpler than another and for showing that an element is an

immediate successor of another element or a chain of limit length.

Theorem 1 (Ehrlich). Let hA, <,

15 x ∈ A and (L, R) be subsets of A for which L < R. Then x = {L | R} if and only if L < {x} < R and {y ∈ A : L < {y} < R} ⊆ {y ∈ A : Ls(x) < {y} < Rs(x)}.

Moreover, x

Theorem 2 (Ehrlich). Let hA, <,

Then y is a right immediate successor of x in A if and only if y = {Ls(x)∪{x} | Rs(x)} and y is a left immediate successor of x if and only if y = {Ls(x) | {x} ∪ Rs(x)}.

Furthermore, y is the immediate successor of the chain (xα)α<β in A of infinite S S limit length if and only if y = { α<β Ls(xα) | α<β Rs(xα)}.

We will later return to binary trees and expound the properties of full binary trees, but first, we must introduce s-hierarchical algebraic structures in order to offer motivation for viewing the surreal numbers as a tree.

3.3 Ordered algebraic structures

One lively area of study pertaining to the surreals is the study of the algebraic structures that initially embed within. These structures, being initial, share the simplicity structure of the surreals, so sums and products may be defined in the same way. Two interesting types of structures, real-closed ordered fields and divis- ible ordered abelian groups are known to be isomorphic to initial substructures of the surreals due to a 2001 publication by Philip Ehrlich [5]. Before moving further, a brief explanation of some more general algebraic structures is in order:

Semigroups: A semigroup is a class which is closed under the operation of addi-

tion (i.e., if x, y are arbitrary members of the semigroup, then x + y is also

a member of the semigroup). All semigroups are associative, that is for all

16 x, y and z,(x + y) + z = x + (y + z). A semigroup is commutative if for any

x and y, x + y = y + x.A monoid is a semigroup which contains an additive

identity element 0 where 0 + x = x + 0 = x for all x. All subsemigroups

of No are commutative and all initial subsemigroups of No are monoids.

Subsemigroups of No also satisfy cancellation laws, that is for all x, y and

z, if x + y = x + z then y = z and if y + x = z + x then y = z.

Groups: A group is a monoid for which every element x has an additive inverse,

denoted −x, for which x+(−x) = (−x)+x = 0. A group in which all elements

commute is called an abelian group. All subgroups of No are abelian groups.

Rings: A ring is an abelian group with the additional operation of multiplication,

which must be associative. Rings are closed under multiplication and are

distributive, that is x·(y +z) = (x·y)+(x·z) and (x+y)·z = (x·z)+(y ·z).

Unital rings are rings which contain a multiplicative identity 1 (where 1·x =

x · 1 = x for all x). Commutative rings are rings in which all elements

commute with respect to multiplication as well as addition. Integral domains

are unital, commutative rings which have no zero divisors, that is, if x·y = 0

then either x = 0 or y = 0. All subrings of No are commutative and have no

zero divisors and all nontrivial initial subrings of No have a multiplicative

identity, making them integral domains.

Fields: A field is an integral domain in which every nonzero element has a mul-

tiplicative inverse, denoted x−1, for which x · x−1 = x−1 · x = 1. A field, F ,

is real-closed if it shares first-order properties with the real numbers. One

of the many equivalent ways to characterize being real-closed is that any

17 polynomial of a single variable p(x) with coefficients in the field satisfies the

intermediate value theorem, that is, for a < b ∈ F and for any u such that

p(a) < u < p(b), there is a c ∈ F where a < c < b and where p(c) = u.

Any of these structures are called ordered if there is a total ordering among the elements which is compatible with the addition operation and (in the case of structures with multiplication) the multiplication operation. That is, for any x, y, z in the structure, if x ≤ y then x + z ≤ y + z and if 0 ≤ x and 0 ≤ y, then

0 ≤ x · y. We also have two more structures which can be constructed over a ring or field:

Modules: A module over a ring (which must be unital) is an abelian group where

elements in the group can be multiplied by elements in the ring (called scalar

multiplication). Scalar multiplication must be compatible with ring mul-

tiplication, compatible with the ring’s multiplicative identity element and

distributive, that is, for x and y in the module and a, b, 1 in the ring,

a(bx) = (a · b)x, 1x = x, a(x + y) = ax + ay and (a + b)x = ax + bx.

Vector Spaces: A vector space is a module over a field.

Adopting the Ehrlich’s terminology in [5], we say that hA, <,

x + y = {xL + y, x + yL | xR + y, x + yR}.

18 We say that hA, <,

We say that hA, <,

R-module and if for all x ∈ R and all y ∈ A, xy = {xLy+xyL −xLyL, xRy+xyR −xRyR | xLy+xyR −xLyR, xRy+xyL −xRyL}.

Substructures of an s-hierarchical ordered algebraic structure will be called initial substructures if they are initial subtrees of the structure. In order to see that all s-hierarchical ordered structures can be initially embedded in No, we must

first establish No as the sole (up to isomorphism) universal s-hierarchical ordered

field, group and vector space.

3.4 Complete s-hierarchical ordered trees and embeddings

In this subsection, we will get an idea of how large No truly is.

Definition 3 (Ehrlich). We say that a lexicographically ordered binary tree hA, <

,

Theorem 4 in [5] establishes that the property of being complete is equivalent

19 to the property of being full, mentioned earlier (each element has exactly two immediate successors and each chain of limit length has exactly one immediate successor). In [2], enough is proved to show that the surreal numbers considered as an s-hierarchical ordered field hNo, <,

Definition 4 (Ehrlich). A mapping f between two lexicographically ordered bi- nary trees A and A0 is said to be an s-hierarchical mapping if for any representation

{L | R}A of any x ∈ A, f(x) = {f(L) | f(R)}A0 where f(L) = {f(y): y ∈ L} and likewise for f(R). Additionally, if A and A0 are s-hierarchical algebraic structures and f is also an embedding (isomorphism) of structures, then f will be said to be an s-hierarchical embedding (isomorphism).

0 0 Lemma 1 (Ehrlich). Let hA, <,

0 and only if for all x, y ∈ A, f(x) < f(y) whenever x < y, f(x)

In [5], Lemma 2, it is shown that s-hierarchical mappings between s-hierarchical ordered algebraic structures are always unique initial embeddings. An s-hierarchical ordered algebraic structure A is said to be universal if every other s-hierarchical ordered structure of the same type can be s-hierarchically embedded into A and

A is said to be maximal if it is not properly contained as a substructure of an s-hierarchical ordered structure of the same type.

Theorem 3 (Ehrlich). Let A be an s-hierarchical ordered group (s-hierarchical ordered field; s-hierarchical ordered vector space). Then A is complete if and only if A is universal if and only if A is maximal if and only if A is isomorphic to No.

20 Of course, this theorem extends to include s-hierarchical ordered monoids, in-

tegral domains, and modules over integral domains, as these are just substructures

of groups, fields and vector spaces. At this point, we have shown that all s-

hierarchical ordered structures are isomorphic to initial substructures of No. Now

we begin work on the task of seeing exactly which ordered structures admit this

s-hierarchical ordering. Before moving on, I will introduce four initial substruc-

tures of the surreals which will be frequently referenced throughout the remainder

of this thesis:

Definition 5. Let N be the set of all surreal numbers x of finite tree-rank for

which Rs(x) = ∅.

Figure 2: No’s natural numbers.

Definition 6. Let Z be the set of all surreal numbers x of finite tree-rank for

which either Ls(x) = ∅ or Rs(x) = ∅.

21 Figure 3: No’s integers.

Definition 7. Let D be the set of all surreal numbers x of finite tree-rank.

Figure 4: No’s dyadic rationals.

Definition 8. Let R be the set of all surreal numbers x for which either (i) x has

finite tree-rank or (ii) x has tree-rank ω and neither Ls(x) nor Rs(x) is finite.

22 Figure 5: No’s real numbers.

As seen in the above figures, these sets are all initial. By Theorem 8 in [5], we know that R is the unique isomorphic copy of the field of real numbers which is an initial subfield of No. Since No is an ordered field, it contains exactly one subring of dyadic rationals, this being D. Throughout, R, D, Z and N will be referred to as the set of real numbers, rational numbers, integers and natural numbers, respec- tively. It should be noted that N is not the unique initial isomorphic copy of the natural numbers considered as an ordered monoid, though it is when the natural numbers are considered as an ordered semidomain (that is, a monoid equipped with an associative, distributive, commutative multiplication operation, a multi- plicative identity element, and no zero divisors). Likewise, Z is not the unique initial isomorphic copy of the integers considered as an ordered group, though it is when the integers are considered as an ordered integral domain.

In order to gain a clear picture of the conditions which must be met by a substructure of No for it to be initial, we must consider some properties of Conway names, canonical proper names which may be assigned uniquely to each surreal number.

23 4 Leaders and Conway Names

4.1 Leaders

Consider an s-hierarchical ordered group A and two elements a, b ∈ A. We say that a is Archimedean equivalent to b (a ≈ b) if there are natural numbers m and n such that m|a| > b and n|b| > a. If this is not the case and a < b, then we say that a is infinitesimal relative to b (a  b). Archimedean equivalence is an equivalence relation and the equivalence classes created by this relation are called Archimedean classes. The positive elements of such a class always form a convex subclass of A: consider a, b and c in A with 0 < a < b < c and [a] = [c] (i.e. the Archimedean class of a is the same as the Archimedean class of c). There is a natural number m such that m · a > c, so m · b > c and, of course, 1 · c > b. Therefore, [b] = [c] as well. By Theorem 1 in [5], every nonempty convex subclass of a lexicographically ordered binary tree contains a simplest member. This simplest member will be called a leader of A. The class of all leaders of A is written Lead(A) and for any x ∈ Lead(A), Lsl(x) = Lead(A) ∩ Ls(x) and Rsl(x) = Lead(A) ∩ Rs(x). The following theorem from [5] gives us three useful properties of leaders:

Theorem 4 (Ehrlich). Let A be a nontrivial s-hierarchical ordered group.

(i) hLead(A),

(ii) If A0 is an initial subgroup of A then Lead(A0) = Lead(A) ∩ A0.

0 0 0 0 (iii) If A is an initial subgroup of A then hLead(A ),

24 In fact, not only is Lead(No) a lexicographically ordered binary tree, but it

is also complete. Recall that in order to do show that Lead(No) is complete, we

must show that for any two subsets L and R of Lead(No) where L < R, there is

an element of Lead(No) between L and R. This is shown in the following lemma

from [5].

Lemma 2 (Ehrlich). Let L and R be subsets of Lead(No) where L < R. Then

Lead(No) 1 No {L | R} exists and equals {0, nL | 2n R} where nL denotes the set of

1 products of elements in L and natural numbers n and likewise for the set 2n R.

By Theorem 3, we have that Lead(No), being as it is a complete lexicograph- ically ordered binary tree, is isomorphic to No. In order to construct this iso- morphism, we make use of Theorem 19 in [2], which states that each nonzero surreal number x is Archimedean equivalent to ωy for some surreal number y. We can then define an s-hierarchical isomorphism, Φ, (called the ω-map by Conway), which sends an element y ∈ No to an element ωy ∈ Lead(No). By Definition 4

y yL 1 yR No No L and Lemma 2, ω = {0, nω | 2n ω } where y = {L | R} , y is a typical element of L and yR is a typical element of R. This result is used in the following

lemma.

y y Lemma 3. (i) For all a, b ∈ R and all y ∈ No, ω · a

y y L y R (ii) For all a = {Ls(a) | Rs(a)} ∈ R − D and all y ∈ No, ω · a = {ω · a | ω · a }

L R where a and a range over the elements of Ls(a) and Rs(a).

L Proof. Proof of (i): Let a = {Ls(a) | Rs(a)} and y = {Ls(y) | Rs(y)}. Let a

R L R range over the elements of Ls(a) and likewise for a , y and y . By Lemma 2,

y yL 1 yR ω = {nω | 2n ω } and by Conway’s definition of multiplication, we have that

25 ωy · a = {L | R} where

L L 1 R 1 R L = {nωy · a + ωy · aL − nωy · aL, ωy · a + ωy · aR − ωy · aR} 2n 2n and

L L 1 R 1 R R = {nωy · a + ωy · aR − nωy · aR, ωy · a + ωy · aL − ωy · aL}. 2n 2n

By Theorem 1, {y ∈ A : L < {y} < R} ⊆ {y ∈ A : Ls(x) < {y} < Rs(x)} and

y y x

by showing that ωy · b also fits into the cut {L | R}.

We first must show that ωy · b > nωyL · a + ωy · aL − nωyL · aL which can be done

by showing ωy · b − nωyL · a − ωy · aL + nωyL · aL = ωy · (b − aL) − nωyL · (a − aL) > 0.

Since (b − aL) and (a − aL) are both positive real numbers and ωy  ωyL , we see

that ωy · (b − aL) > nωyL · (a − aL). Analogous arguments can be used on the other

three types of members of L and R to show that L < ωy · b < R.

Proof of (ii): We know that ωy · a = {L | R} where L and R are defined as in

the proof of (i). We must now show that ωy · aL is cofinal with L and that ωy · aR

is coinitial with R. Consider the elements of L of the form ωy · aL + nωyL · (a − aL).

Clearly, ωy · aL is cofinal with the set of these elements, as ωy  ωyL and so for all

0 0 L y 0 y L yL L y L a ∈ Ls(a) where a > a , ω ·a > ω ·a +nω ·(a−a ). We can show that ω ·a

y R 1 yR R is also cofinal with the set of these elements of the form ω · a − 2n ω · (a − a),

R 0 as this is equivalent to the statement: for any a ∈ Rs(a) there is an a ∈ Ls(a) such

y R 0 1 yR R R 0 that ω · (a − a ) < 2n ω · (a − a). Since y − y > 0, we need only find an a

R 0 (aR−a) (aR−a) 0 such that a − a < 2n . As 2n ∈ R, a ∈ D and D is dense in R, this can

26 be done for any aR and any n. Analogous arguments show that ωy · aR is coinitial

with R.

Remark. This result also follows immediately from Theorems 3.13 and 3.16 [7],

y by considering a number c ∈ R − D such that b = c or b

y subchain of the form (ω ·cn)n<ω where c is the immediate successor of (cn)n<ω. As

y y a ∈ (cn)n<ω and either b ∈ (cn)n<ω or b = c, it is the case that ω · a ∈ (ω · cn)n<ω

y y y y y y and either ω · b ∈ (ω · cn)n<ω or ω · b = ω · c. In either case, ω · a

The second claim can be proved by simply defining a cut in this cofinal subchain.

This lemma will not come into play in our main theorems, but it is useful in its own right. The next lemma, however, will be of use in our main theorem on groups.

1 y x Lemma 4. For any two surreal numbers x and y, if y ∈ Rs(x) then 2n ω

Proof. Suppose without loss of generality that x is the immediate left successor of

y for, if it is not, then the immediate left successor of y is in Rs(x) and if we show

1 y that 2n ω is simpler than the immediate left successor of y for each natural number

1 y x n, then we have shown that 2n ω is simpler than x as well. Consider prNo(ω ), the chain of predecessors of ωx. Clearly, this is a chain of limit length, as ωx is

the immediate successor of another surreal number if and only if x = 0, in which

x case Rs(x) = ∅. By Theorem 13 in [7], we may conclude that prNo(ω ) contains a

y cofinal chain of the form (ω · an)n<ω where (an)n<ω is an initial chain in hR,

27 look like, but by following the proof of the theorem, it is easy to see that there

must be a cofinal chain of this form). We may now use Theorem 16 in [7] to see

y x 1 that in order for the successor of the chain (ω · an)n<ω to be ω , an must equal 2n

x 1 y x for each natural number n. As this chain is contained in prNo(ω ), 2n ω

The introduction of Lead(No) is the first step in assigning canonical proper

names to the surreal numbers.

4.2 Conway names

We may view No as an s-hierarchical vector space over R by considering No as an

s-hierarchical ordered group with scalar multiplication between No and R defined as it is in 3. Any element of this vector space can be formally expressed as a finite

or transfinite power series, called a normal form by Conway, which we shall call

a Conway name. These power series are constructed by Conway in the following

way:

Consider a surreal number x. As mentioned in the previous subsection, either

y0 x = 0 or it is Archimedean equivalent to ω for some y0. If x = 0, then the Conway

name of x is simply 0. Otherwise, consider the sets L = {r ∈ R : ωy0 · r ≤ x} and

L = {r ∈ R : ωy0 · r > x}. We claim that both L and R are nonempty and prove

this claim by first handling the case that x > 0. If x ≈ ωy0 , then there is a natural

y0 1 number n such that x · n > ω , so n ∈ L. There is also a natural number m such

that ωy0 · m > x, so m ∈ R. If x < 0, then consider L0 and R0 defined for −x and let L = {−r : r ∈ R0} and R = {−r : r ∈ L0}. As both L and R are nonempty

28 disjoint sets of real numbers and every real number is in either L or R, L and R

form a partition of the real numbers. As the real numbers are Dedekind-complete,

either L has a greatest element or R has a least element (but not both). Call this

y0 element r0 and write x = ω · r0 + x1, where |x1|  |x|. If x1 = 0, then the

y0 Conway name of x is ω · r0. Otherwise, we repeat the above process with x1 and

y1 write x1 = ω · r1 + x2, where |x2|  |x1|. Continuing in this way, we have that

y0 y1 y2 x = ω · r0 + ω · r1 + ω · r2 + .... If any of these terms is 0, we may stop there,

but this process can continue for any (ordinal) number of steps. Suppose that for

yα some ordinal β (which may or may not be a limit ordinal), we have defined ω · rα

P yα for all α < β then we may define the formal sum α<β ω · rα to be the simplest

yα number x where for all α < β, the α-term of x is equal to ω · rα [2].

Definition 9 (Conway, Ehrlich). The formal expression

X yα ω · rα α<β will be refered to as the Conway name of a surreal number x if either x = 0 and

β = 0 or if x is the simplest number where for all α < β, the α-term of x is equal

yα P yα to ω · rα. If α<β ω · rα is the Conway name of x, then we may treat it as a

P yα proper name for x and write x = α<β ω · rα.

P yα Every surreal number has a distinct Conway name [2]. The expression α<β ω · rα is the Conway name of a surreal number if and only if (yα)α<β∈On is a possibly empty descending sequence of surreal numbers and rα ∈ R − {0} for each α < β.

P yα More specifically, α<β ω · rα is the Conway name of the simplest member of

P yα yα { α<β ω ·rα +a} where |a|  ω for all α < β [2]. The following two theorems,

29 Theorems 15 and 16 in [5], show how we may represent surreal numbers by left

and right sets of Conway names and how simplicity, addition and multiplication

arise when considering the Conway names of surreal numbers.

P yα P yα Theorem 5 (Ehrlich). (i) α<µ ω · rα

(  1  X yα X yα yβ ω · rα = ω · rα + ω · rβ − 2n α<β+1 α<β )No X  1  ωyα · r + ωyβ · r + ; α β 2n α<β 0

(iii) Limit ordinal case

(  1  X yα X yα yβ ω · rα = ω · rα + ω · rµ − 2n α<β α<µ )No X  1  ωyα · r + ωyβ · r + , α µ 2n α<µ 0

The inequalities 0 < n < ω and µ < β indicate that n and µ range over all positive

integers and all ordinals less than β, respectively.

Theorem 6 (Ehrlich). hNo, +H , ·H ,

are understood to be inserted and deleted as needed:

X y X y X y ω · ay +H ω · by = ω · (ay + by), y∈No y∈No y∈No

30  

X y X y X y  X  ω · ay ·H ω · by = ω ·  aµbν ,   y∈No y∈No y∈No (µ,ν)∈No×No µ+H ν=y

P y P y y∈No ω · ay some x ∈ No and ax < bx.

Remark. Ehrlich chose to use the subscript H for these relations in honor of Hans

Hahn, since these definitions of sums, products and ordering are defined `ala Hahn for his ordered fields of formal power series.

4.3 Ordered groups of formal power series

By representing surreal numbers by their Conway names, we can draw parallels between the ordered group of surreal numbers and ordered groups of power series

R(Γ)On.

Definition 10. Let R(Γ)On be the ordered group of power series consisting of all

P yα formal power series of the form α<β rαt where (yα)α<β∈On is a possibly empty descending sequence of elements of an ordered class Γ and rα ∈ R − {0} for each α < β.

When Γ is an ordered group, R(Γ)On forms a field and when Γ is an ordered monoid, R(Γ)On forms an integral domain. Note that we have a very clear iso- morphism between No and R(No)On (established independently by Alling in [1]

P yα and by Ehrlich in [4]). In this isomorphism, the surreal number α<β ω · rα is

P yα simply mapped to α<β rαt . An immediate consequence of this is that every initial ordered structure of No is isomorphic to a substructure of R(No)On. We can now introduce three definitions pertaining to R(Γ)On which will be important in our efforts to categorize the initial substructures of No.

31 Definition 11. An element x ∈ R(Γ)On is said to be a proper truncation of

P yα P yα α<β rαt ∈ R(Γ)On if x = α<σ rαt for some σ < β.

Definition 12. A subfield (subdomain, subgroup) A of R(Γ)On will be said to be truncation closed if every proper truncation of every member of A is itself a member of A.

Definition 13. A subfield (subdomain, subgroup) A of R(Γ)On will be said to be cross sectional if {ty : y ∈ Γ} ⊆ A.

P yα P yα Recall statement (i) in Theorem 5, that α<µ ω ·rα

µ

than x. An immediate ramification this fact follows:

Corollary 1. Initial substructures of R(No)On are truncation closed.

Recall the s-hierarchical isomorphism, Φ : No → Lead(No), from the begin-

ning of this section, where Φ(y) = ωy. As s-hierarchical isomorphisms preserve

simplicity, we can see that for surreal numbers x and y, x

x y ω

Corollary 2. The class of exponents of an initial substructure of R(No)On forms an initial subclass of No.

P yα Suppose some subgroup G of R(Γ)On is truncation closed. If α<β rαt ∈ G,

yα then |rα|t is also in G for each α < β, as G is closed under both truncation

and subtraction. The simple fact that leaders are the simplest members of their

respective Archimedean classes leads to the following:

32 Corollary 3. Initial subgroups of R(No)On are cross sectional.

This is, of course true for initial subdomains, subfields and submodules of

R(No)On as well. However, this is not necessarily true for initial submonoids of

R(No)On, as monoids are not necessarily closed under subtraction. These three corollaries will come into play in categorizing initial groups and integral domains of No and they provide us with one direction of the main result from 2001, categorizing which ordered fields are isomorphic to initial subfields of

No:

Theorem 7 (Ehrlich). An ordered field is isomorphic to an initial subfield of No if and only if it is isomorphic to a truncation closed, cross sectional subfield of a power series field R(Γ)On where Γ is isomorphic to an initial subgroup of No.

As this characterization relies on Γ being an initial subgroup, we have ample motivation to explore precisely which groups can be embedded as initial subgroups of No.

5 Initial Groups

5.1 S-hierarchical ordered modules over integral domains

As Conway names emerge from considering No as an s-hierarchical ordered vec- tor space over the reals, it is only fitting that we gather some results about s- hierarchical ordered vector spaces and, more generally, s-hierarchical ordered mod- ules over integral domains. First, I will introduce a more general definition of what it means for an algebraic structure to be generated by a generating class.

33 Definition 14. Let A be an algebraic structure and let B be a subclass of A.

We say that B generates a substructure B0 ⊆ A if every element in B0 can be constructed by taking finite combinations of algebraic operations on elements of

0 B. This is sometimes written (B)A = B .

Generating classes are not unique, meaning we can have two different generating

classes B1 and B2 which both generate the same structure B.

Example. The group Z of integers is generated by the set {1}. The integral

1 domain D of dyadic rationals is generated by { 2 } but, considered as a group, D is

1 generated by { 2n : n is a natural number}.

Clearly, for an initial subtree A of a lexicographically ordered binary tree A0,

if (L, R) is a partition of A and b = {L | R}A0 , then A ∪ {b} is an initial subtree

of A0 as well. Theorem 6 in [5] provides us with an even more powerful result,

that if A is an initial subspace of an s-hierarchical ordered vector space A0 over a

field K,(L, R) is a partition of A and b = {L | R}A0 , then the ordered subspace

0 0 of A generated by A ∪ {b}, written (A ∪ {b})A0 , is an initial subspace of A . This theorem can be generalized for our purposes as follows, with the proof essentially unchanged from Ehrlich’s:

Theorem 8. Let M 0 be an s-hierarchical ordered module over an integral domain

D and let M be an initial submodule of M 0. If (L, R) is a partition of M and

M 0 0 b = {L | R} , then (M ∪ {b})M 0 is an initial submodule of M .

34 5.2 Preliminary information on groups

As mentioned in Section 3, we will be dealing only with abelian ordered groups.

An ordered group hG, <, +, 0i is said to be discretely ordered if there is an element

a ∈ G such that 0 < a and there is no b ∈ G such that 0 < b < a. The group

hG, <, +, 0i is said to be densely ordered if for any a, b ∈ G where a < b there is a

c ∈ G such that a < c < b.

We can easily show that an initial subgroup G of No is densely ordered if and

only if it is not discretely ordered. If G is densely ordered, then it clearly cannot

be discretely ordered. If G is not densely ordered, then there are elements a, b ∈ G

where a < b but where there is no c ∈ G such that a < c < b. Suppose towards

contradiction that there is no least positive element of G and consider b − a ∈ G.

As there is no least positive element of G and b − a > 0, there must be a g ∈ G

such that 0 < g < b − a, but then a < g + a < b, contradicting our assumption

that there is no element of G between a and b.

We can make an additional statement about densely ordered groups:

Proposition 1. An initial subgroup G of No is discretely ordered if and only if

1 −α there is a g ∈ G of the form 2n ω (where n is a nonnegative integer and α is an ordinal) with no left immediate successor.

1 −α Proof. Note that elements of No of the form 2n ω are precisely those which have only 0 as a left predecessor. We must show that for any element g ∈ G, g is the least positive element of G if and only if Ls(g) = {0} and g has no left immediate successor.

First, suppose there is a least positive element g of G for which Ls(g) 6= {0}.

35 If 0 ∈/ Ls(g), then g is not positive, a contradiction. If there is an a ∈ Ls(g) where

a 6= 0, then a is a positive element less than g, another contradiction.

Next, notice that any least positive element g of G must have no immediate left successor as, if g has a left immediate successor, then this successor is a positive element less than g.

In order to show the other direction we must make use of the assumption that G is initial. Assume that g has no left immediate successor, that Ls(g) = {0} and that there is a number a ∈ G where 0 < a < g. We know that G is lexicographically ordered, so we can say that for any x, y ∈ G where x < y, x is incomparable with y if and only if they have a common predecessor z such that x < z < y [5]. Clearly, a must be incomparable with g, as the only number less than g and comparable with g is zero. Therefore, they must have a common predecessor z such that a < z < g, but by the assumption that Ls(g) = {0}, z must equal 0 and a must be negative, a contradiction.

We must now categorize the initial Archimedean subgroups of No, as our theorem requires that certain subgroups of No be initial. Note that all initial

Archimedean subgroups of No are subgroups of No’s real numbers, R.

z Lemma 5. The groups { 2m : z ∈ Z} for some m ∈ N and groups containing D are

the only Archimedean ordered groups which are initial subgroups of R.

Proof. Suppose that we have an initial subgroup G of R which does not contain

z D. Then there is a greatest m ∈ N such that 2m ∈ G for some z ∈ Z. As G is initial, for all k ≤ m there are integers zk such that we have elements of the

zk form 2k ∈ G. Using closure under subtraction and the fact that we must always

36 have 1 and, therefore Z in any nontrivial initial subgroup of No, we can see that

1 z 2m ∈ G and, by closure under addition and subtraction, { 2m : z ∈ Z} ⊆ G. By

z the assumption that m is the greatest number for which 2m ∈ G, for some z ∈ Z, this is, in fact, an equality.

Note that there are Archimedean groups which are not initial subgroups of R.

1 1 For example, consider the group generated by the set { 3 }. Obviously, 3 itself is in

1 1 this group. For this group to be initial, 2 must be an element of the group as 2

1 1 is a predecessor of 3 . However, there is no way to generate the element 2 through

1 any finite combination of the operations of addition or subtraction on the set { 3 }.

5.3 The Initial subgroups of No

Recall the ordered group of power series, R(Γ)On, from Section 4. For a truncation closed, cross sectional subgroup G of this group, we have Archimedean groups of real numbers corresponding with each exponent y in the class of exponents, Γ:

Definition 15. For a truncation closed, cross sectional subgroup G of R(Γ)On,

y the ordered Archimedean group Ry = {r ∈ R : rt ∈ G} is called the y-coefficient group of G.

We may use the familiar fact that all abelian groups may be considered as a module over the integers to define a class of generators for a subgroup of R(Γ)On:

Proposition 2. If G is a cross sectional, truncation closed subgroup of R(Γ)On and let

( ) X yµ Z = rµt ∈ G : ν is an infinite limit ordinal and r0 = 1 µ<ν

37 y y then Z ∪ {t : y ∈ Γ} ∪ {rt : y ∈ Γ, r ∈ Ry − Z} constitutes a class of generators for G considered as a module over the integers.

This proposition follows immediately from our definitions. Finally, we are pre-

pared to prove a more general theorem about which ordered groups can be embed-

ded as initial subgroups of No.

Theorem 9. An ordered abelian group is isomorphic to an initial subgroup of No

if and only if:

(i) It is isomorphic to a truncation closed, cross sectional subgroup G of a power

series group R(Γ)On where Γ is isomorphic to an initial subclass of No.

(ii) Every y-coefficient group of G is an initial ordered subgroup of R.

(iii) For any x, y ∈ Γ where y ∈ Rs(x), D ⊆ Ry.

Proof. Beginning with the “only if” claim, suppose an ordered group is isomorphic

to an initial subgroup of No. Clearly, this group must also be isomorphic to a

subgroup G of a power series group R(Γ)On. By Corollaries 1, 2 and 3 from Section 4, condition (i) is met.

That condition (ii) is met follows immediately from part (i) of Lemma 3, which

y y states that for all a, b ∈ R and all y ∈ No, ω · a

To show that (iii) is satisfied, consider x, y ∈ Γ where y ∈ Rs(x). As G is cross

x y 1 y x sectional, ω and ω ∈ G. It follows from Lemma 4 that 2n ω

1 nonnegative integer n. Therefore, 2n ∈ Ry for each n and, since groups are closed under addition and subtraction, D ⊆ Ry. We now turn to the “if” portion of the proof. Much of this portion borrows

from Ehrlich’s 2001 proof of Theorem 18 in [5]. The primary difference between

38 these two proofs is that Ehrlich was able to treat an ordered subfield F of R(Γ)On as a vector space over the Archimedean ordered field {r : rt0 ∈ F } whereas we

are only able to treat the group G as a module over the integers. Because of this

limitation, we must show that for each y ∈ Γ, the predecessors of every element in

y the set {ω · r : r ∈ Ry − Z} are in G. This is done in the last few paragraphs of this proof.

Let G be a subgroup of a power series group R(Γ)On meeting conditions (i), (ii) and (iii) above. We can injectively map G to No by mapping each element

P yα P yα α<β rαt ∈ G to α<β ω · rα ∈ No. The image of this mapping, A, is an ordered subgroup of No. We must now show that A is an initial subgroup of No by showing that hA,

We will prove this claim by induction on Γ. Let a0, . . . , aα,... (α < β) be a well-ordering of Γ such that ρNo(aµ) ≤ ρNo(aν) whenever µ < ν < β. Consider

A as a module over the integers. Let Aα be the submodule of A containing 0 as well as all of the elements in A with exponents only from Γα = {aδ : δ ≤ α}. S We see that, considered as a module over the integers, A = α<β Aα. Notice also that since a0 = 0, A0 = R0, which, by condition (ii), is an initial subgroup of

No. Therefore, hA0,

39 let

( ) X yµ [ Zα = ω · rµ ∈ Aα − Aµ : ν is an infinite limit ordinal and r0 = 1 . µ<ν µ<α

aα aα and let b0, . . . , bσ,... (σ < τ) be a well-ordering of Zα ∪ {ω } ∪ {ω · rα : rα ∈

Raα − Z} where for all γ, δ < τ:

aα 1. b0 = ω

aα 2. If bγ ∈ Zα and bδ ∈ {ω · rα : rα ∈ Raα − Z}, then bγ < bδ

3. If bγ, bδ ∈ Zα and the initial sequence of ordinals over which the exponents

in bγ are indexed is contained in the initial sequence of ordinals over which

the exponents in bδ are indexed, then bγ < bδ

aα 4. If bγ, bδ ∈ {ω · rα : rα ∈ Raα − Z}, then bγ < bδ only if ρNo(bγ) ≤ ρNo(bδ).

aα aα By Proposition 2, we can see that Zα ∪ {ω } ∪ {ω · rα : rα ∈ Raα − Z} ∪

S aα µ<α Aµ is a class of generators for Aα. Let bγ be the least member of {ω ·  S  rα : rα ∈ Raα − Z} in our well-ordering. Let B0 = {b0} ∪ µ<α Aµ and let A S
S Bσ = {bσ} ∪ δ<σ Bδ A for 0 < σ < τ. Note that Aα = σ<τ Bσ. Note also that S for σ < γ, bσ ∈/ δ<σ Bδ. We may show that Aα is an initial subtree of No by showing that Bσ is an initial subtree of No for each σ < τ.

aα First, we have b0 = ω . As Γ is assumed to be an initial class, both Ls(aα) and

L Rs(aα) ⊆ {aδ : δ < α}. Let a be a typical element of Ls(aα) and likewise for aR. It follows from Lemma 2 that

  L 1 R b = ωaα = 0, nωa | ωa 0 2n

40 S where n ranges over the natural numbers. As every element of µ<α Aµ − {0} is

aδ S Archimedean equivalent to a unique member of {ω : δ < α} ⊆ µ<α Aµ − {0},

S aL there must be a unique partition (Lα,Rα) of µ<α Aµ where Lα < Rα, {0, nω }

0 1 aR 0 L S is cofinal with Lα and { 2n ω } is coinitial with Rα. Since each a is in µ<α Aµ,

aL 1 aR R {nω } ⊆ Lα. We must also show that { 2n ω } ⊆ Rα, given that each a is in

S R µ<α Aµ. We may do this by making use of assumption (iii). Each a ∈ Rs(aα),

1 aR so D ⊆ RaR . Therefore { 2n ω } ⊆ Rα. By Theorem 1, b0 = {Lα | Rα} and by

Theorem 8, B0 is an initial subtree of No.

We now have two cases, either Zα is empty or it is nonempty. If it is empty,

aα then we move on to the members of {ω · rα : rα ∈ Raα − Z}. If not, then we S suppose as our induction hypothesis that for bσ ∈ Zα, δ<σ Bδ is an initial subtree of No and set out to show that Bσ is an initial subtree of No as well. As bσ ∈ Zα,

P yα bσ has a Conway name of the form α<π ω ·rα where π is an infinite limit ordinal and r0 = 1. By Theorem 5 part (iii), bσ = {L | R} where

( ) X  1  L = ωyα · r + ωyµ · r − α µ 2n α<µ 0

and ( ) X  1  R = ωyα · r + ωyµ · r + α µ 2n α<µ 0

aα S Finally, we have two more cases, either {ω · rα : rα ∈ Raα − Z} − δ<γ Bδ is empty or it is not. If it is empty, then we have shown that Aα is an initial subtree

41 and we are finished. If it is not, then we set out to show that Bσ is an initial subtree of No for γ ≤ σ < τ.

Using our well-ordering and the assumption (ii) that Raα is an initial subset

aα aα of No, we may show that if bσ = ω · rα for some rα ∈ Raα − Z, then {ω · r :

aα S r ∈ Ls(rα)} and {ω · r : r ∈ Rs(rα) are both subsets of δ<σ Bδ. S S Suppose that δ<σ Bδ has been shown to be initial and that bσ 6∈ δ<σ Bδ. Let

L R rα range over Ls(rα), rα range over Rs(rα) and n range over the natural numbers. Consider the following two cases:

L (i) Suppose Ls(aα) is nonempty. Let aα range over Ls(aα). By Lemma 5.8 in [9], we have that

L L aα L aα aα R aα bσ = {ω · rα + ω · n | ω · rα − ω · n}.

We may use the fact that our group is cross sectional and its class of exponents, Γ, is initial and well-ordered in such a way that elements with lower tree-rank precede

L L those with greater tree-rank to see that ωaα and, therefore, each ωaα · n is already S in δ<σ Bδ. Whether Ls(aα) is empty or not, we may use the well-ordering of the

aα aα L aα R S members of {ω · rα : rα ∈ Raα − Z} to see that ω · rα , ω · rα ∈ δ<σ Bδ.

(ii) Suppose Ls(aα) is empty. Again, by Lemma 5.8 in [9], we have that

aα L aα R bσ = {ω · rα | ω · rα }

aα L aα R S and again, know that ω · rα , ω · rα ∈ δ<σ Bδ. S In either case, both the left and right sets for bσ are subsets of δ<σ Bδ. There

0 0 S 0 0 is therefore a partition (Lσ,Rσ) of δ<σ Bδ such that bσ = {Lσ | Rσ}. By Theorem

8, Bσ is an initial subtree of No.

42 Note that at each stage of the final portion of the proof, there may be elements S bσ which are already elements of δ<σ Bδ. Therefore, it is necessary that as part of our induction step, we selected the least bσ (with respect to our well-ordering) S which is not an element in δ<σ Bδ. By exhausting the remaining members of

aα {ω · rα : rα ∈ Raα − Z}, we have that Bσ is an initial subtree of No for all σ < τ, S so for each α < β, Aα is an initial subtree of No. As A = α<β Aα, hA,

6 Initial Integral Domains

6.1 Initial subdomains of No

Now we set out to generalize Ehrlich’s 2001 results on fields to include initial or-

dered integral domains. First we must categorize which types of initial Archimedean

ordered domains we might encounter, as we did for groups. Again, note that all

initial Archimedean subdomains of No are also initial subdomains of R.

Lemma 6. The integral domain Z and domains containing D are precisely the

Archimedean ordered integral domains which are initial subdomains of R.

Proof. Suppose that an initial integral domain D is not isomorphic to Z. Then, it

1 must extend Z and contain some element of the form z + 2 where z is an integer.

1 1 By subtracting z, we may show that 2 ∈ D. As a domain, D is generated by 2 , so

D ⊆ D. If D contains D, then it must be initial, as D ⊆ R and every predecessor of a member of R is a member of D. Clearly, if D = Z, then D is also initial.

Recall that the ordered group of power series, R(Γ)On forms a domain when

43 Γ is an ordered monoid. For a truncation closed, cross sectional subdomain D of

R(Γ)On, we can consider its Archimedean y-coefficient domains:

Definition 16. For a truncation closed, cross sectional subdomain D of R(Γ)On,

y the ordered Archimedean domain Ry = {r ∈ R : rt ∈ D} is called the y-coefficient domain of D.

We may now use the fact that all domains are, in fact, groups with the ad- ditional operation of multiplication to state a more general theorem about which ordered integral domains are isomorphic to initial subdomains of No.

Theorem 10. An ordered integral domain is isomorphic to an initial subdomain of No if and only if:

(i) It is isomorphic to a truncation closed, cross sectional subdomain D of a power series domain R(Γ)On where Γ is isomorphic to an initial submonoid of No.

(ii) Every y-coefficient domain of D is an initial ordered subdomain of R.

(iii) For any x, y ∈ Γ where y ∈ Rs(x), D ⊆ Ry.

Proof. First note that the above conditions are precisely the conditions for initial groups, with the exception of the stipulation that Γ must be form a monoid. This is a necessary condition as, if the image of the domain D is cross sectional, then

for all x, y ∈ Γ, tx and ty are in D, so tx · ty = tx+y is in D and x + y is in Γ.

Of course, the rest of the conditions are necessary for integral domains as well, as

they are necessary for abelian groups. In order to show that the conditions are

also sufficient, we may treat D as a module over Z and repeat the second part of the proof of Theorem 9.

44 An ordered integral domain is said to be discrete if its multiplicative identity,

1 is its least positive element. It is said to be dense if between any two elements there is another element in the domain.

Corollary 4. A densely ordered integral domain is isomorphic to an initial sub- domain of No if and only if it is isomorphic to a truncation closed, cross sectional subdomain D of a power series domain R(Γ)On where Γ is isomorphic to an initial submonoid of No and D is an initial subdomain of D.

Proof. We need only show that D is an initial subdomain of D if and only if conditions (ii) and (iii) are satisfied. Suppose that D is an initial subdomain of

D. This means that D ⊆ R0. If D is cross sectional, then for all y ∈ Γ, R0 ⊆ Ry,

y y since t ∈ D so for all r ∈ R0, r·t ∈ D and r ∈ Ry. This clearly implies that (iii) is met and it implies that (ii) is met as well, as all Archimedean domains containing

D are initial, by Lemma 6. Now suppose that D is not an initial subdomain of D. We must show that condition (ii), condition (iii) and the assumption that the domain is discrete cannot be simultaneously satisfied. If condition (ii) is to be met, then R0 = Z. Of course, if 1 is the least positive element in D, then D is discrete, so we must assume that there is an element d where 0 < d < 1. This

0 element cannot be of the form r · t , as R0 = Z, so it must be an element of the

y form r · t for some y < 0. But then 0 ∈ Rs(y) and as D 6⊆ R0, condition (iii) is not met.

Ehrlich’s 2001 result categorizing initial ordered fields is a special case of this more general result categorizing initial ordered integral domains, as any cross sec- tional, truncation closed subfield F of a power series field R(Γ)On where Γ is isomor-

45 phic to an initial subgroup of No contains D as an initial subdomain. In these sub- fields, every y-coefficient field is, in fact, the same and identical to the residue class

field of the subfield, the residue class field being the quotient ring R/m where R is the subring {x ∈ F : ∃n ∈ N, |x| < n} of F and m = {x ∈ F : ∀r ∈ D+, |x| < r} is the maximal ideal of R.

6.2 The omnific integers (Oz)

The omnific integers were introduced by Conway along with the surreal numbers.

They form a canonical discrete ring where for every surreal number x there is an omnific integer z such that |x − z| < 1. Furthermore, No is Oz’s field of fractions, meaning that every surreal number can be represented as the quotient of two omnific integers [2].

Definition 17 (Conway). A surreal number x is an omnific integer if and only if x = {x − 1 | x + 1}

P yα Theorem 11 (Conway). A surreal number with the Conway name α<β ω · rα is an omnific integer if for all α < β, yα ≥ 0 and rα is an integer if yα = 0.

Oz is in fact, the only (up to isomorphism) universal discrete s-hierarchical ordered integral domain, a claim that we may demonstrate by first noting that all s-hierarchical ordered integral domains are isomorphic to initial subdomains of

No and then showing that all discrete initial subdomains of No are also to initial subdomains of Oz.

Theorem 12. An initial subdomain D of No is discretely ordered if and only if it is an initial subdomain of Oz.

46 Proof. Suppose D is an initial subdomain of Oz. As Oz is an initial subdomain

of No, D must be as well. We can show that D is discrete by supposing towards

contradiction that there is an element in D between 0 and 1. Any such element

1 1 1 has 2 as a predecessor and, as D is initial, 2 must then be in D. But 2 is not

1 1 1 in Oz, as { 2 − 1 | 2 + 1} = 0 6= 2 . For the converse, suppose that there is a discretely ordered initial subdomain D of No containing some element a∈ / Oz.

P yα Consider the Conway name of a, that is, α<β ω ·rα where α and β are ordinals,

(yα)α<β∈On is a descending sequence of surreal numbers and rα ∈ R − {0} for each

P yα 0 0 0 0 α < β. Let b = α<β ω · rα where rα = rα if yα > 0, rα = 0 if yα < 0 and rα is largest integer less than rα if yα = 0. Note that b ∈ Oz, so b = {b − 1 | b + 1}. As b − 1 < a < b + 1, b

1 1 a−b ∈ D, but 0 < a−b < 1, so 2

7 Open Problems and Closing Remarks

7.1 Open Problems

The result in Theorem 12, that all discrete initial subdomains of No are also initial

subdomains of Oz, does not generalize to discrete initial subgroups of No. For

z example, consider the group { 2 : z ∈ Z}. As shown in Lemma 5, this group is an

1 initial subgroup of No, but it is not initial in Oz, as it contains 2 . However, this

group is isomorphic to an initial subgroup of Oz, namely Z.

Question 1. Is every discrete initial subgroup of No isomorphic to an initial

subgroup of Oz?

47 Question 1 is the first of two open problems which are asked in this thesis. The

second question is of great importance to our result on initial subdomains of No.

Recall that one of the necessary conditions in Theorem 10 for an ordered integral

domain to be isomorphic to an initial subdomain of No is that it be isomorphic

to a subdomain of the power series domain R(Γ)On where Γ is isomorphic to an initial submonoid of No. Of course, this raises the following question:

Question 2. What are the necessary and sufficient conditions for an ordered

monoid to be isomorphic to an initial submonoid of No?

We have some partial results on this question. First, some necessary conditions:

any initial submonoid of No must be isomorphic to a submonoid S of a power series

group R(Γ)On. By Corollary 1, S must be truncation closed. By Corollary 2, Γ must be isomorphic to an initial subclass of No.

We can state some additional necessary conditions. For example, for any y ∈ Γ,

the Archimedean monoid {r ∈ R : rty ∈ S} must be an initial submonoid of R. However, if S is not cross sectional, these conditions have little import and initial submonoids of No need not necessarily cross sectional, as is demonstrated by the following counterexample:

Example. Let S be the monoid generated by all finite sums of elements in {n} ∪

1 1 {ω} ∪ {ω + 2n } ∪ {ω + ω } where n ranges over the natural numbers. S is an initial

1 submonoid, but it does not contain ω , so it is not cross sectional.

Of course, in order to claim that S is initial in the first place, we need the

following result of Antongiulio Fornasiero [8].

48 Lemma 7 (Fornasiero). If A and B are an initial subclasses of No, then A + B =

{a + b : a ∈ A, b ∈ B} and the additive group generated by A are also initial in

No.

Though not explicitly mentioned, the following corollary is relatively trivial:

Corollary 5. If A is an initial subclass of No then the additive monoid generated by A is an initial submonoid of No.

Proof. Clearly, any union of initial subclasses of No is also an initial subclass of

No. Let A be an initial subclass of No. Let A0 = A and for each natural number n > 0, let An = An−1 + An−1. By Lemma 7, each An is initial in No so long as

0 S An−1 is, and as A0 is initial, all An are initial. Let A = n<ω An. Any sum of

length n of members of A is in An, and therefore, all finite sums of members of

A are in A0, so A0 is the additive monoid generated by A. As A0 is the union of

initial subclasses of A, A0 is initial.

A positively ordered monoid is an ordered monoid for which all non-zero ele-

ments are greater than zero. A negatively ordered monoid is an ordered monoid

for which all non-zero elements are less than zero. If M − is an initial negatively

ordered submonoid of No, then clearly, −M − = {−m : m ∈ M −} is an initial positively ordered submonoid of No. If M + is an initial positively ordered sub- monoid of No and M − is a an initial negatively ordered submonoid of No, then

M + ∪ M − is an initial subclass of No, so by Corollary 5, the monoid M generated by M + ∪M − is also an initial submonoid of No. Therefore, in our search for which monoids are initial, we may focus only on which positively ordered monoids are initial.

49 7.2 Closing remarks

The results established in this thesis successfully expand our knowledge of s- hierarchical ordered algebraic structures, though much still remains in the dark. I believe that Question 1 will be resolved as Philip Ehrlich and I begin work on a paper highlighting and expanding on the results in this thesis. Unfortunately, it seems that Question 2 may have a very complicated answer, complicated enough to negatively impact the usefulness of the result. More work should be done before giving up on the question, but perhaps it would be more useful to find different types sufficient conditions which pick out only (but not all) monoids which are isomorphic to initial submonoids of No. Of course, the monoid case has a direct effect on the question of which semidomains are isomorphic to initial subsemido- mains of No. S-hierarchical ordered semidomains went largely undiscussed in this thesis, but an understanding of these semidomains are another step towards a more general theory of s-hierarchical ordered algebraic structures.

Another important s-hierarchical ordered algebraic structure which went un- classified in this thesis are s-hierarchical ordered modules. The classification of initial subdomains of No in Theorem 10 is a step towards a classification if ini- tial submodules of No, but the connection between initial submodules and initial subdomains is not as strong as the connection between initial subfields and ini- tial subspaces. Theorem 7 in [5], which states that an ordered F -vector space is isomorphic to an initial subspace of No if and only if F is isomorphic to an initial subfield of No has no clear analogue for modules over domains isomorphic to initial subdomains of No. One easy way to see this is to consider that every abelian group is a module over Z, so if this theorem generalized to domains word

50 for word, all abelian groups would be isomorphic to initial subgroups of No, which is certainly not the case. I believe that the clear next step is to determine which conditions a domain D must meet for an ordered D-module to be isomorphic to an initial submodule of No. Work on this question will also provide us with additional information regarding initial subgroups of No.

Of course, the surreals serve as more than a backdrop for this theory of s- hierarchical ordered algebraic structures. Interest in the surreals has grown even within the last year, as Alessandro Beraducci and Vincenzo Mantova have recently completed a paper demonstrating that No admits a derivation satisfying many desirable properties and Ovidiu Costin, Philip Ehrlich and Harvey M. Friedman are currently completing a paper on integration on the surreals. Additionally, the surreals are deeply linked to transseries, a topic which has been greatly advanced in the last year due to work by Matthias Aschenbrenner, Lou van den Dries and

Joris van der Hoeven.

That such a rich structure was created in such a simple manner is truly re- markable. It seems inevitable that the surreals will draw further attention in this upcoming year, especially with a proposal to hold a special session on surreal num- bers at the 2016 Joint Mathematics Meetings recently accepted by the Association for Symbolic Logic. It is my hope to expand on the results in this thesis and on the small but exciting body of work surrounding the surreal numbers in the coming years.

51 References

[1] N. L. Alling. Foundations of Analysis over Surreal Number Fields. North-

Holland Publishing Company, Amsterdam, 1987.

[2] J. H. Conway. On Numbers and Games. Academic Press, 1976.

[3] F. Drake. Set Theory; an Introduction to Large Cardinals. North-Holland

Publishing Co., Amsterdam, 1974.

[4] P. Ehrlich. An alternative construction of Conway’s ordered field No. Algebra

Universalis, 25:7–16, 1988.

[5] P. Ehrlich. Number systems with simplicity hierarchies: A generalization

of Conway’s theory of surreal numbers. The Journal of Symbolic Logic,

66(3):1231–1258, 2001.

[6] P. Ehrlich. The rise of non-Archimedean mathematics and the roots of a miscon-

ception i: The emergence of non-Archimedean systems of magnitudes. Archive

for History of Exact Sciences, 60:1–121, 2006.

[7] P. Ehrlich. Conway names, the simplicity hierarchy and the surreal number

tree. Journal of Logic & Analysis, 3(1):1–26, 2011.

[8] A. Fornasiero. Recursive definitions on surreal numbers. ArXiv Mathematics

e-prints, 2006. http://arxiv.org/abs/math/0612234/.

[9] H. Gonshor. An Introduction to the Theory of Surreal Numbers. Cambridge

University Press, 1986.

52