Truncations of ordered abelian groups

Abstract

We give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a $$+$$ + , but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments $$[0, \tau ]$$ [ 0 , τ ] of an ordered abelian group, with a truncation of the addition. We prove that any model of these axioms (i.e. a truncated ordered abelian group) is an initial segment of an ordered abelian group. We define Presburger TOAG’s, and give a criterion for a TOAG to be a Presburger TOAG, and for two Presburger TOAG’s to be elementarily equivalent, proving analogues of classical results on Presburger arithmetic. Their main interest for us comes from the model theory of certain local rings which are quotients of valuation rings valued in a truncation [0, a] of the ordered group $${\mathbb {Z}}$$ Z or more general ordered abelian groups, via a study of these truncations without reference to the ambient ordered abelian group. The results are used essentially in a forthcoming paper (D’Aquino and Macintyre, The model theory of residue rings of models of Peano Arithmetic: The prime power case, 2021, arXiv:2102.00295) in the solution of a problem of Zilber about the logical complexity of quotient rings, by principal ideals, of nonstandard models of Peano arithmetic.

The model theory of residue rings of models of Peano Arithmetic: The prime power case

“…It is notable in Stage 2 that the M/p k M are Henselian local rings, and models of a natural set of axioms involving TOAGS (see [12]), truncated ordered abelian groups.…”

“…We can construe v as a map to a "truncated ordered abelian group", henceforward called TOAG. In [12] axioms which are true in initial segments of ordered abelian groups are identified. We list them here.…”

“…Some are obvious, while others need some calculations. In [12] it is also shown that models of these axioms can be realized as initial segments of ordered abelian groups. Notice that in [12] it is not specified if the order is discrete.…”

“…In [12] it is also shown that models of these axioms can be realized as initial segments of ordered abelian groups. Notice that in [12] it is not specified if the order is discrete. We are mostly interested in discrete orders and some extra axioms will be added later.…”

“…We expand the language for TOAG with an extra constant symbol 1. The extra axioms are also sufficient as shown in the following theorem in [12]. .…”