Elimination of quantifiers for ordered valuation rings

Abstract

Cherlin and Dickmann [2] proved that the theory RCVR of real closed (valuation) rings admits quantifier-elimination (q.e.) in the language ℒ = {+, −, ·, 0, 1, <, ∣} for ordered rings augmented by the divisibility relation “∣”. The purpose of this paper is to prove a form of converse of this result:Theorem. Let T be a theory of ordered commutative domains (which are not fields), formulated in the language ℒ. In addition we assume that:(1) The symbol “∣” is interpreted as the honest divisibility relation: (2) The following divisibility property holds in T:If T admits q.e. in ℒ, then T = RCVR.We do not know at present whether the restriction imposed by condition (2) can be weakened.The divisibility property (DP) has been considered in the context of ordered valued fields; see [4] for example. It also appears in [2], and has been further studied in Becker [1] from the point of view of model theory. Ordered domains in which (DP) holds are called in [1] convexly ordered valuation rings, for reasons which the proposition below makes clear. The following summarizes the basic properties of these rings:Proposition I [2, Lemma 4]. (1) Let A be a linearly ordered commutative domain. The following are equivalent:(a) A is a convexly ordered valuation ring.(b) Every ideal (or, equivalently, principal ideal) is convex in A.(c) A is a valuation ring convex in its field of fractions quot(A).(d) A is a valuation ring and its maximal ideal MA is convex (in A or, equivalently, in quot (A)).(e) A is a valuation ring and its maximal ideal is bounded by ± 1.

Weakly o-minimal structures and real closed fields

“…It follows from Theorem 5 of [5] that the theory of K * also has elimination of quantifiers. An expansion of the argument in [6] now shows that Th K * is weakly o-minimal, and it can be verified that this theory does not have prime models.…”

“…Dickmann in [6] observes that real closed rings are weakly o-minimal. He also proves that every convexly ordered valuation ring which admits quantifierelimination in the language L Div is a real closed ring.…”

“…We also say that a complete theory T is weakly ominimal if all its models are weakly o-minimal. Weak o-minimality was introduced by Dickmann in [6], where some algebraic examples are developed and some basic facts about weakly o-minimal ordered groups and rings are obtained.…”

Imaginaries in real closed valued fields

“…(y 1 , y 2 ) := (x 1 .y 1 − x 2 .y 2 , x 1 .y 2 + x 2 .y 1 ). We also define V := {(x 1 , x 2 ) ∈ K 2 : v(x 2 1 + x 2 2 ) ≥ 0}. Now we can define K as the L R+V -structure with domain K 2 in the obvious way.…”

Minimality conditions on circularly ordered structures

“…Real closed fields with a proper convex valuation ring [13] provide an important example, and the theory was developed in [4,6,7,16,17,21].…”