## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 33, Number 1 (2009), 95-129.

### Quantifier Elimination for Lexicographic Products of Ordered Abelian Groups

Shingo Ibuka, Hirotaka Kikyo, and Hiroshi Tanaka

#### Abstract

Let $\Lg = \{+,-,0\}$ be the language of the abelian groups, $L$ an expansion of $\Lg(<)$ by relations and constants, and $\Lmod = \Lg \cup \{\equiv_n\}_{n \geq 2}$ where each $\equiv_n$ is defined as follows: $x \equiv_n y$ if and only if $n|x-y$. Let $H$ be a structure for $L$ such that $H|\Lg(<)$ is a totally ordered abelian group and $K$ a totally ordered abelian group. We consider a product interpretation of $H \times K$ with a new predicate $I$ for $\{0\}\times K$ defined by N.~Suzuki \cite{Sz}.

Suppose that $H$ admits quantifier elimination in $L$.

- 1. If $K$ is a Presburger arithmetic with smallest positive element $1_K$ then the product interpretation $G$ of $H \times K$ with a new predicate $I$ admits quantifier elimination in $L(I, 1) \cup \Lmod$ with $1^G = (0^H, 1_K)$.
- 2. If $K$ is dense regular and $K/nK$ is finite for every integer $n \geq 2$ then the product interpretation $G$ of $H \times K$ with a new predicate $I$ admits quantifier elimination in $L(I, D) \cup \Lmod$ for some set $D$ of constant symbols where $G \models I(d)$ for each $d \in D$.
- 3. If $K$ admits quantifier elimination in $\Lmod(<, D)$ for some set $D$ of constant symbols then the product interpretation $G$ of $H \times K$ with a new predicate $I$ admits quantifier elimination in $L(I, D) \cup \Lmod$ unless $K$ is dense regular with $K/nK$ being infinite for some $n$.

Conversely, if the product interpretation $G$ of $H \times K$ with a new predicate $I$ admits quantifier elimination in $L(I, D) \cup \Lmod$ for some set $D$ of constant symbols such that $G \models I(d)$ for each $d \in D$ then $H$ admits quantifier elimination in $L \cup \Lmod$, and $K$ admits quantifier elimination in $\Lmod(<, D)$.

We also discuss the axiomatization of the theory of the product interpretation of $H \times K$. %For some set $C$ of constants in $K$.

#### Article information

**Source**

Tsukuba J. Math., Volume 33, Number 1 (2009), 95-129.

**Dates**

First available in Project Euclid: 1 September 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1251833209

**Digital Object Identifier**

doi:10.21099/tkbjm/1251833209

**Mathematical Reviews number (MathSciNet)**

MR2553840

**Zentralblatt MATH identifier**

1188.03022

**Subjects**

Primary: 03C10: Quantifier elimination, model completeness and related topics 03C64: Model theory of ordered structures; o-minimality 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces [See also 46A40]

**Keywords**

ordered abelian groups quantifier elimination

#### Citation

Ibuka, Shingo; Kikyo, Hirotaka; Tanaka, Hiroshi. Quantifier Elimination for Lexicographic Products of Ordered Abelian Groups. Tsukuba J. Math. 33 (2009), no. 1, 95--129. doi:10.21099/tkbjm/1251833209. https://projecteuclid.org/euclid.tkbjm/1251833209