Quantifier elimination results for products of ordered Abelian groups
Nobuya Suzuki · Tsukuba Journal of Mathematics · 2004
IntroductionKomori [1] introduced the notion of semi-discrete ordered Abelian group with divisible infinitesimals.Roughly speaking, such groups are products of a Z-like group and a Q-like group.In [1], he showed that such groups are axiomatized by his set $SC$ of axioms.In fact he showed that $SC$ is complete and admits quantifier elimination (QE) in some language expanding $L_{og}=$ $\{0, +, -, <\}$ .In this paper, we shall evolve his study and prove QE for products of ordered Abelian groups $H$ and $K$ , where $H$ admits QE and $K$ is divisible.However, like him, we need to expand the language slightly.First let us explain Komori's axiom.$SC$ is the following set of sentences:1. the axioms for ordered Abelian groups; 2. the axioms for a semi-discrete ordering $0<1$ , $\forall x(2x<1\vee 1<2x)$ ; 3. the axioms for infinitesimals$\forall x(2x<1\rightarrow nx<1)$ $(n=2,3, \ldots)$ ; 4. the axioms for $D_{n}\prime s$ $\forall x(D_{n}(x)\leftrightarrow\exists y\exists z(-1<2z<1\wedge x=ny+z)$ $(n=2,3, \ldots)$ $\forall x(D_{n}(x)\vee D_{n}(x+1)\vee\cdots\vee D_{n}(x+n-1))$ $(n=2,3, \ldots)$ ; 5. the axioms for divisible infinitesimals $\forall x(-1<2x<1\rightarrow\exists y(x=ny)$ $(n=2,3, \ldots)$ ; 6. the axiom for existence of infinitesimals $\exists x(0<x<1)$ ;