Editing Surreal number (section)

==Alternative realizations==

Alternative approaches to the surreal numbers complement the original exposition by Conway in terms of games.

===Sign expansion===

====Definitions====

In what is now called the ''sign-expansion'' or ''sign-sequence'' of a surreal number, a surreal number is a [[function (mathematics)|function]] whose [[domain of a function|domain]] is an [[ordinal number|ordinal]] and whose [[codomain]] is {{math|{{mset| −1, +1 }}}}.<ref name=G1986 />{{rp|at=ch. 2}} This notion has been introduced by Conway himself in the equivalent formulation of L-R sequences.<ref name="Con01" />

Define the binary predicate "simpler than" on numbers by: {{mvar|x}} is simpler than {{mvar|y}} if {{mvar|x}} is a [[subset|proper subset]] of {{mvar|y}}, i.e. if {{math|dom(''x'') < {{if mobile|<br />|}}dom(''y'')}} and {{math|1=''x''(''α'') = ''y''(''α'')}} for all {{math|''α'' < dom(''x'')}}.

For surreal numbers define the binary relation {{math|<}} to be lexicographic order (with the convention that "undefined values" are greater than {{math|−1}} and less than {{math|1}}). So {{math|''x'' < ''y''}} if one of the following holds:
* {{mvar|x}} is simpler than {{mvar|y}} and {{math|1=''y''(dom(''x'')) = +1}};
* {{mvar|y}} is simpler than {{mvar|x}} and {{math|1=''x''(dom(''y'')) = −1}};
* there exists a number {{mvar|z}} such that {{mvar|z}} is simpler than {{mvar|x}}, {{mvar|z}} is simpler than {{mvar|y}}, {{math|1=''x''(dom(''z'')) = −1}} and {{math|1=''y''(dom(''z'')) = +1}}.

Equivalently, let {{math|1=''δ''(''x'',{{hsp}}''y'') = min({ dom(''x''), dom(''y'')} ∪ { ''α'' :{{if mobile|<br />|}}
''α'' < dom(''x'') ∧ ''α'' < dom(''y'') ∧ ''x''(''α'') ≠ ''y''(''α'') })}},
so that {{math|1=''x'' = ''y''}} if and only if {{math|1=''δ''(''x'',{{hsp}}''y'') = dom(''x'') = dom(''y'')}}.  Then, for numbers {{mvar|x}} and {{mvar|y}}, {{math|1=''x'' < ''y''}} if and only if one of the following holds:
* {{nowrap|{{math|1=''δ''(''x'',{{hsp}}''y'') = dom(''x'') ∧ ''δ''(''x'',{{hsp}}''y'') < dom(''y'') ∧ ''y''(''δ''(''x'',{{hsp}}''y'')) = +1}};}}
* {{nowrap|{{math|1=''δ''(''x'',{{hsp}}''y'') < dom(''x'') ∧ ''δ''(''x'',{{hsp}}''y'') = dom(''y'') ∧ ''x''(''δ''(''x'',{{hsp}}''y'')) = −1}};}}
* {{nowrap|{{math|1=''δ''(''x'',{{hsp}}''y'') < dom(''x'') ∧ ''δ''(''x'',{{hsp}}''y'') < dom(''y'') ∧ ''x''(''δ''(''x'',{{hsp}}''y'')) = −1 ∧ ''y''(''δ''(''x'',{{hsp}}''y'')) = +1}}.}}

For numbers {{mvar|x}} and {{mvar|y}}, {{math|''x'' ≤ ''y''}} if and only if {{math|1=''x'' < ''y'' ∨ ''x'' = ''y''}}, and {{math|1=''x'' > ''y''}} if and only if {{math|''y'' < ''x''}}. Also {{math|''x'' ≥ ''y''}} if and only if {{math|''y'' ≤ ''x''}}.

The relation {{math|<}} is [[transitive relation|transitive]], and for all numbers {{mvar|x}} and {{mvar|y}}, exactly one of {{math|''x'' < ''y''}}, {{math|1=''x'' = ''y''}}, {{math|''x'' > ''y''}}, holds (law of [[trichotomy (mathematics)|trichotomy]]).  This means that {{math|<}} is a [[linear order]] (except that {{math|<}} is a proper class).

For sets of numbers {{mvar|L}} and {{mvar|R}} such that {{math|∀''x'' ∈ ''L'' ∀''y'' ∈ {{if mobile|<br />|}}''R'' (''x'' < ''y'')}}, there exists a unique number {{mvar|z}} such that
* {{nowrap|{{math|1=∀''x'' ∈ ''L'' (''x'' < ''z'') ∧ ∀''y'' ∈ ''R'' (''z'' < ''y'')}},}}
* For any number {{mvar|w}} such that {{math|∀''x'' ∈ ''L'' (''x'' < ''w'') ∧ ∀''y'' ∈ {{if mobile|<br />|}}''R'' (''w'' < ''y'')}}, {{math|1=''w'' = ''z''}} or {{mvar|z}} is simpler than {{mvar|w}}.

Furthermore, {{mvar|z}} is constructible from {{mvar|L}} and {{mvar|R}} by transfinite induction. {{mvar|z}} is the simplest number between {{mvar|L}} and {{mvar|R}}. Let the unique number {{mvar|z}} be denoted by {{math|''σ''(''L'',{{px2}}''R'')}}.

For a number {{mvar|x}}, define its left set {{math|''L''(''x'')}} and right set {{math|''R''(''x'')}} by
* {{nowrap|{{math|1=''L''(''x'') = {{mset|1= ''x''{{!}}{{sub|''α''}} : ''α'' < dom(''x'') ∧ ''x''(''α'') = +1 }}}};}}
* {{nowrap|{{math|1=''R''(''x'') = {{mset|1= ''x''{{!}}{{sub|''α''}} : ''α'' < dom(''x'') ∧ ''x''(''α'') = −1 }}}},}}

then {{math|1=''σ''(''L''(''x''),{{hsp}}''R''(''x'')) = ''x''}}.

One advantage of this alternative realization is that equality is identity, not an inductively defined relation. Unlike Conway's original realization of the surreal numbers, however, the sign-expansion requires a prior construction of the ordinals, while in Conway's realization, the ordinals are constructed as particular cases of surreals.

However, similar definitions can be made that eliminate the need for prior construction of the ordinals. For instance, we could let the surreals be the (recursively-defined) class of functions whose domain is a subset of the surreals satisfying the transitivity rule {{math|1=∀''g'' ∈ dom ''f'' (∀''h'' ∈ dom ''g'' (''h'' ∈ dom ''f'' ))}} and whose range is {{math|{{mset| −, + }}}}. "Simpler than" is very simply defined now: {{mvar|x}} is simpler than {{mvar|y}} if {{math|''x'' ∈ dom ''y''}}. The total ordering is defined by considering {{mvar|x}} and {{mvar|y}} as sets of ordered pairs (as a function is normally defined):  Either {{math|1=''x'' = ''y''}},  or else the surreal number {{math|1=''z'' = ''x'' ∩ ''y''}} is in the domain of {{mvar|x}} or the domain of {{mvar|y}} (or both, but in this case the signs must disagree). We then have {{math|''x'' < ''y''}} if {{math|1=''x''(''z'') = −}} or {{math|1=''y''(''z'') = +}} (or both).  Converting these functions into sign sequences is a straightforward task; arrange the elements of {{math|dom ''f''}}{{hsp}} in order of simplicity (i.e., inclusion), and then write down the signs that {{math|''f''}} assigns to each of these elements in order.  The ordinals then occur naturally as those surreal numbers whose range is {{math|{{mset| + }}}}.

====Addition and multiplication====

The sum {{math|''x'' + ''y''}} of two numbers {{mvar|x}} and {{mvar|y}} is defined by induction on {{math|dom(''x'')}} and {{math|dom(''y'')}} by {{math|1=''x'' + ''y'' = ''σ''(''L'',{{px2}}''R'')}}, where
* {{math|1=''L'' = {{mset| ''u'' + ''y'' : ''u'' ∈ ''L''(''x'') }} ∪ {{mset| ''x'' + ''v'' : ''v'' ∈ ''L''(''y'') }}}},
* {{math|1=''R'' = {{mset| ''u'' + ''y'' : ''u'' ∈ ''R''(''x'') }} ∪ {{mset| ''x'' + ''v'' : ''v'' ∈ ''R''(''y'') }}}}.

The additive identity is given by the number {{math|1=0 = {{(}} {{)}}}}, i.e. the number {{math|0}} is the unique function whose domain is the ordinal {{math|0}}, and the additive inverse of the number {{mvar|x}} is the number {{math|−''x''}}, given by {{math|1=dom(−''x'') = dom(''x'')}}, and, for {{math|''α'' < dom(''x'')}}, {{math|1=(−''x'')(''α'') = −1}} if {{math|1=''x''(''α'') = +1}}, and {{math|1=(−''x'')(''α'') = +1}} if {{math|1=''x''(''α'') = −1}}.

It follows that a number {{mvar|x}} is [[Positive number|positive]] if and only if {{math|1=0 < dom(''x'')}} and {{math|1=''x''(0) = +1}}, and {{mvar|x}} is [[negative number|negative]] if and only if {{math|1=0 < dom(''x'')}} and {{math|1=''x''(0) = −1}}.

The product {{mvar|xy}} of two numbers,  {{mvar|x}} and {{mvar|y}}, is defined by induction on {{math|dom(''x'')}} and {{math|dom(''y'')}} by {{math|1=''xy'' = ''σ''(''L'',{{px2}}''R'')}}, where
* {{math|1=''L'' = {{mset| ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''L''(''x''), ''v'' ∈ ''L''(''y'') }} ∪ {{mset| ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''R''(''x''), ''v'' ∈ ''R''(''y'') }}}}
* {{math|1=''R'' = {{mset| ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''L''(''x''), ''v'' ∈ ''R''(''y'') }} ∪ {{mset| ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''R''(''x''), ''v'' ∈ ''L''(''y'') }}}}

The multiplicative identity is given by the number {{math|1=1 = {{mset| (0, +1) }}}}, i.e. the number {{math|1}} has domain equal to the ordinal {{math|1}}, and {{math|1=1(0) = +1}}.

====Correspondence with Conway's realization====

The map from Conway's realization to sign expansions is given by {{math|1=''f''{{hsp}}({{mset| ''L'' {{!}} ''R'' }}) = ''σ''(''M'',{{px2}}''S'')}}, where {{math|1=''M'' = {{mset| ''f''{{hsp}}(''x'') : ''x'' ∈ ''L'' }}}} and {{math|1=''S'' = {{mset| ''f''{{hsp}}(''x'') : ''x'' ∈ ''R'' }}}}.

The [[inverse map]] from the alternative realization to Conway's realization is given by {{math|1=''g''(''x'') = {{mset| ''L'' {{!}} ''R'' }}}}, where {{math|1=''L'' = {{mset| ''g''(''y'') : ''y'' ∈ ''L''(''x'') }}}} and {{math|1=''R'' = {{mset| ''g''(''y'') : ''y'' ∈ ''R''(''x'') }}}}.

===Axiomatic approach===

In another approach to the surreals, given by Alling,<ref name="Alling" />  explicit construction is bypassed altogether. Instead, a set of axioms is given that any particular approach to the surreals must satisfy. Much like the [[Real numbers#Axiomatic approach|axiomatic approach]] to the reals, these axioms guarantee uniqueness [[up to]] isomorphism.

A triple <math display=inline>\langle \mathbb{No}, \mathrm{<}, b \rangle</math> is a surreal number system if and only if the following hold:

* {{math|<}} is a [[total order]] over <math display=inline>\mathbb{No}</math>
* {{mvar|b}} is a function from <math display=inline>\mathbb{No}</math> [[onto]] the class of all ordinals ({{mvar|b}} is called the "birthday function" on <math display=inline>\mathbb{No}</math>).
* Let {{mvar|A}} and {{mvar|B}} be subsets of <math display=inline>\mathbb{No}</math> such that for all {{math|''x'' ∈ ''A''}} and {{math|''y'' ∈ ''B''}}, {{math|''x'' < ''y''}} (using Alling's terminology, {{math|〈 ''A'', ''B'' 〉}} is a "Conway cut" of <math display=inline>\mathbb{No}</math>). Then there exists a unique <math display=inline>z \in \mathbb{No}</math> such that {{math|''b''(''z'')}} is minimal and for all {{math|''x'' ∈ ''A''}} and all {{math|''y'' ∈ ''B''}}, {{math|''x'' < ''z'' < ''y''}}. (This axiom is often referred to as "Conway's Simplicity Theorem".)
* Furthermore, if an ordinal {{mvar|α}} is greater than {{math|''b''(''x'')}} for all {{math|''x'' ∈ ''A'', ''B''}}, then {{math|''b''(''z'') ≤ ''α''}}. (Alling calls a system that satisfies this axiom a "full surreal number system".)

Both Conway's original construction and the sign-expansion construction of surreals satisfy these axioms.

Given these axioms, Alling<ref name="Alling"/> derives Conway's original definition of {{math|≤}} and develops surreal arithmetic.

===Simplicity hierarchy===

A construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip Ehrlich.<ref name="Ehr12">{{cite journal
 |author      = Philip Ehrlich
 |author-link  = Philip Ehrlich
 |year        = 2012
 |title       = The absolute arithmetic continuum and the unification of all numbers great and small
 |journal     = The Bulletin of Symbolic Logic
 |volume      = 18
 |issue       = 1
 |pages       = 1–45
 |url         = http://www.ohio.edu/people/ehrlich/Unification.pdf
 |access-date  = 2017-06-08
 |doi         = 10.2178/bsl/1327328438
 |s2cid = 18683932
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20171007095144/http://www.ohio.edu/people/ehrlich/Unification.pdf
 |archive-date = 2017-10-07
}}</ref> The difference from the usual definition of a tree is that the set of ancestors of a vertex is [[well-order]]ed, but may not have a [[maximal element]] (immediate predecessor); in other words the order type of that set is a general ordinal number, not just a natural number. This construction fulfills Alling's axioms as well and can easily be mapped to the sign-sequence representation. Ehrlich additionally constructed an isomorphism between Conway's maximal surreal number field and the maximal [[hyperreal field|hyperreals]] in [[von Neumann–Bernays–Gödel set theory]].<ref name="Ehr12" /> 

===Hahn series===

Alling<ref name="Alling" />{{rp|at=th. 6.55, p. 246}} also proves that the field of surreal numbers is isomorphic (as an ordered field) to the field of [[Hahn series]] with real coefficients on the value group of surreal numbers themselves (the series representation corresponding to the normal form of a surreal number, as defined [[#Powers of ω and the Conway normal form|above]]). This provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory.

This isomorphism makes the surreal numbers into a [[valued field]] where the valuation is the additive inverse of the exponent of the leading term in the Conway normal form, e.g., {{math|1=''ν''(''ω'') = −1}}. The [[valuation ring]] then consists of the finite surreal numbers (numbers with a real and/or an infinitesimal part). The reason for the sign inversion is that the exponents in the Conway normal form constitute a reverse well-ordered set, whereas Hahn series are formulated in terms of (non-reversed) well-ordered subsets of the value group.