Presburger Arithmetic and Pseudo-Recursive Saturation
David Llewellyn-Jones School of Mathematics and StatisticsThe University of Birmingham
Edgbaston, Birmingham, B15 2TT, U.K. Email
http://www.flypig.co.uk August 2nd, 2001 The first half of this thesis looks at well known general properties of Presburger arithmetic, including quantifier elimination, types, compactness and homogeneity. It is accessible to the algebraist as well as the model theorist. Let
be a model of Presburger arithmetic. Define the residue map 
sending an element to the sequence of its remainders and the standard part 
for 
to be the supremum of the set 
. Define a partitioning of our model by the equivalence relation 
if and only if 
and let 
be the natural valuation.
We say that 
is pseudo-recursively saturated if: the inverse image of the residue map is dense in 
; for 
there exists 
such that 
; and 
forms a dense linear order with least point 
and no greatest point.
We prove that pseudo-recursive saturation implies homogeneity and give results in the opposite direction indicating an affinity between the two.
Our main result concerns the automorphism group, 
, of the countable pseudo-recursively saturated models of Presburger arithmetic, giving a correlation between the closed normal subgroups of 
and sets of tuples of the standard parts of the model. We define 
to be stQ-closed if: all subsets of 
defined to be those tuples of a certain length form a group; if 
and 
then 
; and similarly if 
then there exists some 
such that 
. We then have that 
is a closed normal subgroup of 
if and only if 
preserves some stQ-closed set 
or 
.
From this we are able to provide some results about the closed normal subgroups of 
and to present a pair of Galois connections between closed normal subgroups of 
, stQ-closed subsets of the set of standard parts and equivalence relations on 
.