Editing Surreal number (section)

===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'') }}}}.