GENERIC EXPANSIONS OF GEOMETRIC THEORIES

Convention

We use capital letters $M, N, P,K$ for the $\mathcal {L}$ -structures and $ A, B, C,D$ and $X, Y, Z$ , show finite and infinite sets, respectively. Instead of $X\cup Y$ we would write $XY$ . For tuples $\bar {a}, \bar {b}$ in a model of M of T (or even $T^{\forall }$ ) by the quantifier-free type of $\bar {a}$ over $\bar {b}$ , denoted by $\operatorname {\mathrm {qftp}}_{\mathcal {L}}(\bar {a}/\bar {b})$ , we mean the set of all quantifier-free formulas $\varphi (\bar {x},\bar {b})$ such that $M\models \varphi (\bar {a},\bar {b})$ .

The dimension obtained by $\operatorname {\mathrm {acl}}$ in T is denoted by $\dim $ . So for a set Y, $\dim (\bar {a}/Y)$ is the size of the $\operatorname {\mathrm {acl}}$ -base of $\{a_1,\ldots ,a_n\}$ over Y. If $\bar {b}=(b_{1},\ldots ,b_{n})$ , then $\dim (\bar {a}/\bar {b})=\dim (\bar {a}/\{b_1,\dots ,b_m\})$ , for $\bar {a}=(a_{1},\ldots ,a_{n})$ . If $\varphi (\bar {x},\bar {y})$ is an $\mathcal {L}$ -formula and $\bar {b}\in M^{|\bar {y}|}$ , then $\dim (\varphi (M,\bar {b}))=\max \{ \dim (\bar {a}/\bar {b}):\ M\models \varphi (\bar {a},\bar {b})\}$ .

The set Y is called $\dim $ -independent from Z over X and denoted by if for every $\bar {a}\in Y^{|\bar {a}|}$ , $ \dim (\bar {a}/X) =\dim (\bar {a}/XZ)$ . Moreover, if $Y\cap Z=X$ we state that Y and Z are free over X. To emphasize that Y and Z are free over X, the set $YZ$ is written as $Y\oplus _{X} Z$ .

In the following fact, some properties of the dimension and $\dim $ -independence are expressed. As a convention, assume that all subsets and tuples are from (a sufficiently saturated) model M of T.

By the above properties, one can prove the following lemma. The first part of the following statement appears in [Reference Chatzidakis and Pillay11, Lemma 2.3] and the second part in [Reference Jalili, Pourmahdian and Khani14, Lemma 2.2]. This lemma is later used in Section 3.

The following definition characterizes the notion of a strong formula. Intuitively, such formula is called strong, since it has enough information about $\dim $ .

Particularly, the second item of the above definition deduces $\dim (\bar {b}^{\prime }_{1}/\bar {a}^{\prime })\geq \dim (\bar {b}_1/\bar {a})$ .

The $\bigvee $ -Definability in Fact 2.1 proves that a strong formula exists for every pair of distinct finite sequences. The proof of the following lemma can be found in [Reference Jalili, Pourmahdian and Khani14, Lemma 2.6].

The following fact presents fundamental properties of $\dim $ -independence.

Recall that an $\mathcal {L}$ -embedding $f:M\to N$ is algebraically closed if $\operatorname {\mathrm {acl}}(f(M))=f(M)$ .

The following observation expresses a property of the free amalgamation which can be easily proved. Assume T has the free amalgamation property and a model M of T which is $\lambda $ -saturated, for $\lambda \geq \aleph _{0}$ .

In the rest of this section, the notion of indecomposability is presented. This notion has already appeared in [Reference Baldwin1] with a different name, federated. While this notion is used in [Reference Jalili, Pourmahdian and Khani14] for the axiomatization of bi-colored expansions, here it is only used in Section 4 to prove Theorem 4.16.

The assumption of indecomposability provides the following desirable property for bases. The proof can be found in [Reference Jalili, Pourmahdian and Khani14, Lemma 2.10].

2.1 Bi-colored expansions

In the following subsection, the theory T is geometric with the quantifier elimination and the free-amalgamation properties. First, we fix some more notation and conventions.

Unary predicate p called the coloring predicate is added to the language $\mathcal {L}$ and we denote the new language by $\mathcal {L}_{p}$ . Also, the class $\mathcal {K}$ is defined as the class of all $\mathcal {L}_{p}$ -structures whose universe M is a model of $T^{\forall }$ . For every $X\subseteq M$ , the $\mathcal {L}_{p}$ -structure generated by X in M is denoted by $\langle X \rangle _{M}$ , or $\langle X\rangle $ . Therefore, M is finitely generated if there is a finite set $A\subseteq M$ such that $M=\langle A\rangle $ . As a convention, also $\emptyset $ is a finitely generated structure in $ \mathcal {K}$ . The element x is called colored if $M\models p(x)$ . If there is no ambiguity we may write $p(x)$ instead of $M\models p(x)$ . Moreover, $p(\bar {x})$ is used instead of $\bigwedge _{i=1}^{n}p(x_{i})$ , when $\bar {x}=(x_1,\ldots ,x_n)$ . Additionally $p(X/Y)$ denotes the set of colored elements of $X\setminus Y$ . Throughout this paper, by $\operatorname {\mathrm {tp}}(X/Y)$ we mean the type of X over Y in $\mathcal {L}_{p}$ . Moreover for the rest of the paper, a rational number $0<\alpha \leq 1$ is fixed. Then for every structure $M\in \mathcal {K}$ and a finite subset A of M, a pre-dimension map $\delta _{\alpha }$ is defined as

$$ \begin{align*} \delta_{\alpha}(A)=\dim(A)-\alpha|p(A)|. \end{align*} $$

Furthermore, for every $X\subseteq M$ we define $\delta _{\alpha }(A/X)=\dim (A/X)-\alpha |p(A/X)|$ .

Note that by the quantifier elimination, $\delta _\alpha (A)$ is independent from the choice of M. For any finite subsets $A, B$ and C it is not hard to check that

$$ \begin{align*} \delta_{\alpha}(AB/C)=\delta_{\alpha}(A/BC)+\delta_{\alpha}(B/C). \end{align*} $$

The pre-dimension $\delta _a$ is submodular, i.e.,

$$\begin{align*}\delta_{\alpha}(AB)+\delta_{\alpha}(A\cap B)\leq \delta_{\alpha}(A)+\delta_{\alpha}(B).\end{align*}$$

For the class $\mathcal {K}$ , one can define the subclass $\mathcal {K}_{\alpha }^{+}$ as

$$ \begin{align*} \mathcal{K}_{\alpha}^{+}:=\{M\in \mathcal{K}| \delta_{\alpha}(A)\geq 0 \text{for every} A\subseteq_{\text{fin}}M\}. \end{align*} $$

For simplicity, if $M\in \mathcal {K}_{\alpha }^{+}$ then for every $X\subseteq M$ , we say $X\in \mathcal {K}_{\alpha }^{+}$ if $\langle X\rangle \in \mathcal {K}_{\alpha }^{+}$ . Therefore, an embedding $f:X\to Y$ means the embedding $\hat {f}:\langle X\rangle \to \langle Y\rangle $ where $X,Y\in \mathcal {K}_{\alpha }^{+}$ . In the rest of the paper, every structure is assumed to be in $\mathcal {K}_{\alpha }^{+}$ , unless we emphasize otherwise.

Clearly, by the previous conventions $\emptyset \in \mathcal {K}^+_\alpha $ . Moreover, $\bigvee $ -Definability implies that $\mathcal {K}_{\alpha }^{+}$ is axiomatizable in $\mathcal {L}_{p}$ by the $\mathcal {L}_{p}$ -sentences

$$ \begin{align*} \neg \exists x_1,\ldots, x_n\big(\psi_l(x_1,\dots,x_n)\wedge \bigvee _{\substack{Y\subseteq \{x_1,\dots,x_n\} \\ |Y|\geq k}} p(Y)\big), \end{align*} $$

where $ \psi _l $ asserts that the dimension of $(x_1,\ldots ,x_l)$ is bounded by l, and $l<\alpha k$ .

In the following definition, the notion of closedness is presented.

One can relax the first item of the above definition by taking arbitrary finite subsets A and B of X, since $\delta _{\alpha }(B/A)=\delta _{\alpha }(AB/A)$ . Hence the first item is a special case of item 2.

For simplicity, we will omit $\alpha $ in $\leqslant _{\alpha }$ and $\delta _{\alpha }$ .

Note that $\alpha>0$ implies that all algebraic points over $\emptyset $ are non-colored. By the definition of $\delta _{\alpha }$ and since $\alpha>0$ whenever $X\leqslant M$ , then $\neg p(x)$ for all $x\in \operatorname {\mathrm {acl}}_{M}(X)\setminus X$ . So, $ \langle X\rangle _M$ and $\operatorname {\mathrm {acl}}_{M}(X)$ are closed in M.

By 2 and 4 of the above remark, one can conclude if $M_1,M_2\leqslant M$ then $M_1\cap M_{2}\leqslant M$ .

Next, we introduce the concept of closure and an intrinsic extension of a set.

The following statements are well-known facts about basic properties of $\operatorname {\mathrm {cl}}_M$ .

Since $\alpha $ is rational, the values of $\delta $ take place in a discrete set. Hence $\operatorname {\mathrm {cl}}(A)$ is finite for each finite set A. Therefore $\operatorname {\mathrm {cl}}(A)\subseteq \operatorname {\mathrm {acl}}(A)$ , where $\operatorname {\mathrm {acl}}$ denotes the algebraic closure in the sense of $\mathcal {L}_{p}$ .

The following definition separates two different types of closed extensions.

3.1 Axiomatization of rich structures

As we mentioned earlier, there exists a complete theory $\mathbb {T}_{\alpha }$ that axiomatizes the class of $\lambda $ -rich structures of $(\mathcal {K}_{\alpha }^{+},\leqslant )$ . The following notion of intrinsic formulas is utilized in this axiomatization.

Let $A\leqslant B $ and $ B $ is a transcendental extension of A. Set

Respectively for any $C\in \mathcal {C}_{A,B}$ let $\mathcal {F}_{C}$ be the collection of the formulas $\varphi (\bar {x},\bar {y};\bar {z})\in \Delta _{A,B;C}$ which also $\dim $ -forks. Note that by strong finite character of Fact 2.5 the set $\mathcal {F}_{C}$ is non-empty.

Below for an $\mathcal {L}$ -formula $\varphi (\bar {x}, \bar {y})$ , let $\varphi ^{*}(\bar {y},\bar {x}):= \varphi (\bar {x}, \bar {y})$ . Respectively, for a natural number k, consider $D_{\varphi ^{*}, k}(\bar {x})$ and $d_{\varphi ^{*}, k}(\bar {x})$ as the formulas introduced in Lemma 2.2 and Fact 2.1.

Also, for $A\subseteq B$ we denote $\bar {a}$ and $\bar {b}$ to be enumerations for A and $B\setminus A $ , respectively.

The above items can be expressed as first-order axiom-schemes; the second item is mentioned before Definition 2.12.

Proof By Theorem 3.6, $\mathbb {M}$ is $\aleph _{0}$ -saturated as a model of T. Now we just prove that the third item in Definition 3.8 holds. Consider $\bar {a},\bar {b}$ , $\varphi $ , $\Phi _{0}$ and $\mathcal {C}_{A,B}$ as in the hypothesis of item $3$ . Let $\bar {a}^{\prime }$ in $\mathbb {M}$ be such that $\mathbb {M}\models D_{\varphi ^{\ast },k}(\bar {a}^{\prime })\wedge d_{\varphi {\ast },k}(\bar {a}^{\prime })$ . So there is $\bar {b}^{\prime }\in \mathbb {M}^{|\bar {b}|}$ with $\mathbb {M}\models \varphi (\bar {b}^{\prime },\bar {a}^{\prime })$ , $\bar {b}^{\prime }\cap \operatorname {\mathrm {acl}}(\bar {a}^{\prime })=\emptyset $ , and $\dim (\bar {b}^{\prime }/\bar {a}^{\prime })=k$ . Since $ \operatorname {\mathrm {cl}}(\bar {a}^{\prime })$ is finite, by $\omega $ -saturation of $\mathbb {M}$ and Observation 2.8 we may assume that $\langle \operatorname {\mathrm {cl}}(\bar {a}^{\prime })\rangle \cap \bar {b}^{\prime }=\emptyset $ and $ \bar {b}' $ and $ \langle \operatorname {\mathrm {cl}}(\bar {a}') \rangle $ are free over $ \bar {a} '$ . Consider the $ \mathcal {L} $ -structure generated by $ \operatorname {\mathrm {cl}}(\bar {a}')\bar {b}' $ in $ \mathbb {M} $ and make it into an $ \mathcal {L}_p $ -structure $ N $ by coloring $ \operatorname {\mathrm {cl}}(\bar {a}')\bar {b}' $ with the same colors as $\operatorname {\mathrm {cl}}(\bar {a}) \bar {b} $ and leaving the rest of the elements non-colored. It is clear that $\langle \operatorname {\mathrm {cl}}(\bar {a}')\rangle \leqslant \mathbb {M}$ . We claim that also $\langle \operatorname {\mathrm {cl}}(\bar {a}')\rangle \leqslant {N}$ . By this claim, since $\mathbb {M}$ is $\aleph _{0}$ -rich, there is a strong embedding $f:N\to \mathbb {M}$ which fixes $\langle \operatorname {\mathrm {cl}}(\bar {a}^{\prime })\rangle $ pointwise. Let $f(\bar {b}^{\prime })=\bar {e}$ . Then

$$ \begin{align*} \mathbb{M}\models \big{(}\varphi(\bar{a}^{\prime},\bar{e})\wedge \bigwedge _{p(b_i)}p(e_{i})\wedge \bigwedge_{\neg p(b_i)}\neg p(e_i)\big{)}. \end{align*} $$

Furthermore we prove that

$$ \begin{align*} \mathbb{M}\models \neg\exists\bar{z} \psi_{A,B,C}(\bar{a}^{\prime},\bar{e};\bar{z}), \end{align*} $$

for every $C\in \Phi _{0}$ . Otherwise suppose there exists $\bar {c}^{\prime }$ in $\mathbb {M}$ such that

$$ \begin{align*} \mathbb{M}\models \psi_{A,B,C}(\bar{a}^{\prime},\bar{e};\bar{c}^{\prime}), \end{align*} $$

for some $C\in \Phi _{0}$ . As $\bar {c}^{\prime }$ is disjoint from $\bar {a}^{\prime }$ and $\bar {e}$ , it follows that $\bar {c}^{\prime }\subseteq \operatorname {\mathrm {cl}}(\bar {a}^{\prime }\bar {e})\setminus \bar {a}^{\prime }\bar {e}$ . Let $A^{\prime }$ and $B^{\prime }$ be the union of elements of tuples $\bar {a}^{\prime }$ and $\bar {a}^{\prime }\bar {e}$ , respectively. Since $ \bar {b}' $ and $ \langle \operatorname {\mathrm {cl}}(\bar {a}') \rangle $ are free over $ \bar {a} '$ it follows that $\operatorname {\mathrm {cl}}(B^{\prime })=\operatorname {\mathrm {cl}}(A^{\prime })\oplus _{A^{\prime }}B^{\prime }$ . As $\psi _{A,B,C}$ is an intrinsic formula, we have that $\bar {c}^{\prime }\subseteq \operatorname {\mathrm {cl}}(B^{\prime })$ and $\bar {c}^{\prime }\cap B^{\prime }=\emptyset $ . Now since , it follows that . But this contradicts the fact that $\psi _{A,B,C}(\bar {x},\bar {y};\bar {z})$ $\dim $ -forks.

Now to prove the claim, let $ \bar {b}^{\prime }_1\subseteq \bar {b}'$ and $ \bar {d}\subseteq N\setminus \operatorname {\mathrm {cl}} (\bar {a}^{\prime })\bar {b}'$ . Notice that implies $\dim (\bar {b}^{\prime }_1/\langle \operatorname {\mathrm {cl}}(\bar {a}^{\prime })\rangle )=\dim (\bar {b}^{\prime }_{1}/\bar {a}^{\prime })$ . Moreover as $\varphi $ is strong, by Definition 2.3 we have $\dim (\bar {b}^{\prime }_{1}/\bar {a}^{\prime })\geq \dim (\bar {b}_{1}/\bar {a})$ . Therefore

$$ \begin{align*} \delta(\bar{b}^{\prime}_1\bar{d}/\langle \operatorname{\mathrm{cl}}(\bar{a}^{\prime})\rangle)&= \dim(\bar{b}^{\prime}_1\bar{d}/\langle \operatorname{\mathrm{cl}}(\bar{a}^{\prime})\rangle)-\alpha |p(\bar{b}^{\prime}_1\bar{d})|\\ &=\dim(\bar{b}^{\prime}_1/\langle \operatorname{\mathrm{cl}}(\bar{a}^{\prime})\rangle)-\alpha |p(\bar{b}^{\prime}_1)|\\ &=\dim(\bar{b}^{\prime}_{1}/\bar{a}^{\prime})-\alpha |p(\bar{b}^{\prime}_1)|\\ &\geq \dim(\bar{b}_{1}/\bar{a})-\alpha |p(\bar{b}_1)|\geq 0.\\[-33pt] \end{align*} $$

In the following, we first prove that any $\aleph _{0}$ -saturated model of $\mathbb {T}_{\alpha }$ is $\aleph _{0}$ -rich. This result, together with the fact that any two $\aleph _{0}$ -rich structures of $\mathcal {K}^{+}_{\alpha }$ are back and forth equivalent, implies that $\mathbb {T}_{\alpha }$ is complete.

4.1 $NTP_{2}$ and strongness of $\mathbb {T}_{\alpha }$

We begin this section by showing that $\mathbb {T}_{\alpha }$ is $NTP_{2}$ (respectively strong), provided that T is $NTP_{2}$ (respectively strong). To this end we show that the burden of $\mathbb {T}_{\alpha }<\infty $ under the assumption that T is $NTP_{2}$ . We first review the basic related concepts and facts all of which can be found in [Reference Ben-Yacov and Chernikov8].

Next we recall the notion of burden. As in [Reference Ben-Yacov and Chernikov8], for notational convenience, we consider an extension Card $^{*}$ of the linear order on cardinals by adding a new maximum element $\infty $ and replacing every limit cardinal $\kappa $ by two new elements $\kappa _{-}$ and $\kappa _{+}$ with $\kappa _{-}<\kappa _{+}$ . The standard embedding of cardinals into Card $^{*}$ identifies $\kappa $ with $\kappa _{+}$ . In the following, whenever we take a supremum of a set of cardinals, we will be computing it in Card $^{*}$ .

For a theory $\Gamma $ put $\operatorname {\mathrm {bdn}}(\Gamma )=\operatorname {\mathrm {bdn}}(\text {"}x=x\text {"})$ .

From now on, as before, we work in the monster model $\mathfrak {C}$ of $\mathbb {T}_{\alpha }$ .