A SHORT PROOF OF NUBLING'S RESULT (Model theoretic techniques for constructing infinite structures)
Ikuo Yoneda · Institutional Repositories DataBase (IRDB) · 2008
Nubling shows that CM-triviality ( $=non-2$-amplenss) is preserved under reducts in finite U-rank theories.We give a short proof. REDUCTION AND INDEPENDENCELet $T^{-}$ be a reduct of $T$ .Let $\mathcal{M}\models T,$ $\mathcal{M}^{-}\models T^{-}$ be big models.$a,$ $b,$ $c,$ $\ldots\overline{a},$ $\overline{b},\overline{c},$ $\ldots$ denote finite tuples, and $A,$ $B,$ $C,$ $\ldots$ denote small sets.Let $A\subset \mathcal{M}^{eq}$ .$ACL^{eq}(A)$ denotes the algebraically closure of $\mathcal{A}$ in $T$ , and acleq(A) denotes the algebraically closure of $A\cap(\mathcal{M}^{-})^{eq}$ .Let $\overline{a}\in(\mathcal{M}^{-})^{eq}$ .TP $(\overline{a}/A)$ denotes the type of $\overline{a}$ over $A$ in $T$ , and tp $(\overline{a}/A)$ denotes the type of $\overline{a}$ over $A$ in $T^{-}$ .SU denotes Lascar rank in $T$ , and su denotes Lascar rank in $T^{-}$ .We show the following fact in the last section.Fact 1.1.Let $T$ be a simple theory having $EHI$ such that $T^{-}$ also has $EHI$ , where $T^{-}$ be a reduct of T. Let $a,$ $C\subset(\mathcal{M}^{-})^{eq}$ and $B\subset \mathcal{M}^{eq}$ .If a $|L_{B}C_{f}$ then a $L_{B^{-}}^{-}C_{f}$ where $B^{-}=ACL^{eq}(B)\cap(\mathcal{M}^{-})^{eq}$ and $\backslash L^{-}$ is the non-forking relation in $T^{-}$ .Proposition 1.2.If SU $(T)<\omega$ , then su $(T^{-})<\omega$ .Proof.Let $a\in(\mathcal{M}^{-})^{eq},$ $A\subset \mathcal{M}^{eq}$ .Put $A^{-}=ACL^{eq}(A)\cap \mathcal{M}^{eq}$ .We will show that there exists $\overline{a}'\models$ tp $(a/A^{-})$ such that SU $(a'/A)\geq$ su $(a'/A^{-})$ by inductionwe see $a_{1}'L_{A}^{-}-B^{-}$ .As $B^{-}\subseteq B$ and a $L_{A^{-}}^{-}B$ , by Fact 1.1, we see $\overline{a}_{1}'L_{A}B$ .Therefore we have SU $(a_{1}'/A)\geq$ SU $(a_{1}'/B)+1\geq$ su $(a_{1}'/B^{-})+1=n+1=$ su $(a/A^{-})=$ su $(a_{1}'/A^{-})$ , as desired.