Ordered groups
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An ordered group is a group 
equipped with a linear
ordering 
with
for all 
.
A unary representation of 
is an embedding 
for
some linearly ordered set 
,
where 
is the
ordering by universal comparison.
Natural example: 
itself
with translations on the left, i.e. 
.
If 
is dense or
, then 
has QE and is o-minimal.
is a
linearly ordered set. Each 
is represented as a series 
.
be an ordered
field. Let 
be an
ordered vector space over 
is an ordered
group for the product 
and the lexicographic ordering.
as
the strictly increasing affine map 
,
and the ordering is then 
iff 
for sufficiently
large 
.
. 
: free product of 
and 
.
for 
lying in extensions
of 
and 
, the extension 
is determined by an ordering
on the quotient group 
where
centralizer
extension
is Abelian. For
, we have 
and 
.
.
, 
.
,
.
.
induces an elementary class in 
of thus orderable groups.
and a convex normal subgroup 
such that 
.
be an o-minimal
structure.
for the set of
germs 
at 
of definable maps 
. We have 
for a natural “asymptotic” structure on 
is closed under
composition 
, and 
is an ordered group.
:
real-closed field. 
:
real exponential field.
, we have 
.
is a divisible
ordered Abelian group, then 
.
, we have 
. Moreover 
. Moreover 
contains
all germs of functions that can be obtained as combinations of
, 
, and semialgebraic functions.
with
. How is the
inequation
, if 
, then 
intuitively grows faster than 
.
for all finite iterations.
,
where
,
we have 
.
and all 
, there is an 
with 
.
is an ordering which is compatible with 
. The relation 
if (
and 
) is a convex equivalence relation.
is said
central if for all 
,
there is a 
such that
.
,
there is a central 
with 
.
expands the real ordered field, then must 
by constructing ordered
differential fields of formal series with composition laws:
with 
. For 
with 
, the derivative
and the compositum 
of formal Laurent series.
is not a group. 
has no inverse
of Puiseux series, and 
is a growth order group.
conjugate in 
?
and 
are not conjugate.
be a linearly
ordered abelian group. Then 
is an ordered field for the Cauchy product
is infinitesimal
then 
.
of
transseries: generalized series involving 
, 
and 
and combinations
thereof.
and 
must be
uniformly bounded.
enjoys a
derivation 
that acts termwise, e.g.
enjoys a
composition law 
that acts termwise on the right:
.
.
, 
.
with
are conjugate.
and 
has a solution 
for all 
,
we deduce that
in which 
solve the conjugacy equation
.
of
. Its growth is
transexponential: 
for
sufficiently large 
.
define a transexponential function?
.
with composition where (
with formal
symbols 
for all 
, where 
and
where for all 
, we have a symbol 
with 
and
is not a group.
where 
is bijective
and strictly increasing.
and composition law 
on 
extend in a
natural way to 
.
is
an ordered group.
, all
conjugacy equations can now be solved:
in 
of surreal
numbers. It comes with a linear ordering 
of
simplicity.
are subsets with 
,
then there is a unique 
with 
.
,
, we have
naturally contains the reals 
with their Hessenberg (commutative) arithmetic.
corresponding to the ordinal 
.
and 
, define 
as the number
for polynomial asymptotics.
.
on 
.
and 
as in 
on surreal numbers.
, and every
surreal number can be represented as a (possibly infinitely
nested) hyperseries in the same vein as elements of 
is a growth order
group with exactly three conjucacy classes.
, 
, we have a unique isomorphism
. Indeed this holds
for 
, and is carried over
by conjugacy.
operation, one
can define, for all strictly 
-decreasing
ordinal indexed sequence 
of “
-simple”
elements of 
, a transfinite composition
.
of Abelian ordered groups 
indexed by a linearly ordered set 
?