FOM: computable ordered nonabelian group
Matthew Frank
mfrank at math.uchicago.edu
Wed Nov 7 01:28:35 EST 2001
In response to my request for a computable ordered nonabelian group, Chris
Miller suggested "the group of infinitely increasing (i.e., $\uparrow
+\infty$) germs at $+\infty$ of (absolutely) semialgebraic functions",
where the group operation is composition and the order is the natural
order. This indeed fits the bill; my unpacking of the suggestion is
below. --Matt
f(x) is semi-algebraic if f(x)=y is equivalent to
exists z1, ... zn such that phi(x,y,z1,...,zn),
where phi is quantifier-free and in the language of fields
So we can represent these functions by formulas in order to discuss
computability.
Our group is the set of semi-algebraic functions such that
f(x) -> infinity when x->+infinity, i.e.
for every N, there exists c, for all x>c, f(x)>N
under the equivalence operation of f(x)~g(x) iff
there is some c such that f(x)=g(x) for all x>c
with the group operation being composition,
and the order being f(x)>g(x) iff
there is some c such that f(x)>g(x) for all x>c.
We have inverses because all these functions are eventually monotone,
and inverses are computable by switching x and y in phi.
Group composition is (well-defined and) computable:
if f(x)=y is equivalent to alpha
and g(y)=z is equivalent to beta
then g(f(x))=z is equivalent to (exists y)(alpha and beta),
which can be rewritten in the desired form.
The order relation is computable by the decidability of RCF.
Trichotomy holds because distinct semi-algebraic functions can be equal
at only finitely many points.
If f>g then hf>hg by the eventual monotonicity of h
If f>g then fh>gh trivially.
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