An $\mathrm{NSOP}_{1}$ theory without the existence axiom

One of the core informal questions of model theory is to determine what role stability theory, famously introduced by Morley ([19]) and Shelah ([23]) to classify the number of non-isomorphic models (of a given size) of a first-order theory, can play in describing theories that are themselves unstable–and that may not even be simple. This project was initiated in large part in celebrated work of Kim ([15]) and Kim and Pillay ([17]), who showed that the (forking-)independence relation has many of the same properties in simple theories that it does in stable theories, and in fact that these properties characterize simplicity. In a key step towards the non-simple case, Kaplan and Ramsey ([12]) then defined Kim-independence over models, which generalizes the definition of forking-independence:

In, for example, [6], [12], [13], [14], it is shown that Kim-independence has many of the same properties in theories without the first strict order property–$\mathrm{NSOP}_{1}$ theories–that forking-independence has in simple theories. It is also shown that $\mathrm{NSOP}_{1}$ theories have a characterization in terms of properties of Kim-independence, similarly to how simplicity has a characterization in terms of properties of forking-independence. However, in contrast to the case of simple theories, this work only describes the properties of a relation $a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}^{K}b$ defined when $M$ is a model, leaving open the question of whether independence phenomena in $\mathrm{NSOP}_{1}$ theories extend from independence over models to independence over sets.

The following axiom of first-order theories, though introduced earlier under different names (see, e.g. [5]), was defined by Dobrowolski, Kim and Ramsey in [9]:

This is equivalent to every type $p\in S(A)$ having a global extension that does not fork over $A$. All simple theories satisfy the existence axiom ([15]), while the circular ordering is an example of a dependent theory not satisfying the existence axiom ([16], Example 2.11). In [9] it is shown that, in $\mathrm{NSOP}_{1}$ theories satisfying the existence axiom, it is possible to extend the definition of Kim-independence $a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{C}^{K}b$ from the case where $C$ is a model to the case where $C$ is an arbitrary set, so that the relation $a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{C}^{K}b$ will have similar properties to Kim-independence over models in general $\mathrm{NSOP}_{1}$ theories. Specifically, because, under the existence axiom, (nonforking-)Morley sequences are defined over arbitrary sets, they can be used in place of invariant Morley sequences¹¹1It is easy to see that invariant Morley sequences are not defined over arbitrary sets in $\mathrm{NSOP}_{1}$ theories: for example, any set $A$ such that $\mathrm{acl}(A)\neq\mathrm{dcl}(A)$. to define Kim-dividing of a formula over a set:

Dobrowolski, Kim and Ramsey ([9]) show that, in an $\mathrm{NSOP}_{1}$ theory satisfying the existence axiom, the Kim-independence relation $\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}^{K}$ as defined above over arbitrary sets satisfies symmetry and the independence theorem (for Lascar strong types), and that Kim-dividing over arbitrary sets satisfies Kim’s lemma and coincides with Kim-forking. Chernikov, Kim and Ramsey, in [2], show even more properties of Kim-independence over arbitrary sets in $\mathrm{NSOP}_{1}$ theories satisfying the existence axiom, including transitivity and witnessing. (All of these properties correspond to the properties of Kim-independence over models proven in general $\mathrm{NSOP}_{1}$ theories in [12], [13].) Motivated by these results, [9], and later [3], ask:

We show that the answer to this question is no:

This contrasts with the work of Kim, Kim and Lee in [3], where, for the definition of Kim-forking over sets given by Dobrowolski, Kim and Ramsey in [9] (Defintion 1.3 above), it is shown that no type $p\in S(A)$ Kim-forks over $A$ (i.e. contains a formula Kim-forking over $A$), where $A$ is an arbitrary set in an $\mathrm{NSOP}_{1}$ theory. In fact, our example is the first known example of a theory without the strict order property, or $\mathrm{NSOP}$ theory, not satisfying the existence axiom. Prior to our results, the theory of an algebraically closed field with a generic multiplicative endomorphism, constructed by d’Elbée in [8], was suggested there as a candidate for an $\mathrm{NSOP}_{1}$ theory without the existence axiom; whether the theory constructed by d’Elbée satisfies the existence axiom was left unresolved in that article. However, as discussed in a personal communication with d’Elbée ([7]), it is expected that that theory, which was shown in [8] to be $\mathrm{NSOP}_{1}$, actually does satisfy the existence axiom.

Our construction is based on the theory of $\omega$-stable free pseudoplanes, a classical example of a non-one-based $\omega$-stable theory with trivial forking discussed in, say, [20]. The theory of $\omega$-stable free pseudoplanes is the theory of undirected graphs of infinite minimum degree without cycles. Variations of this construction appear in e.g. [10], [1], [24], [11]. Note also that some arguments from the below, including the strategy, in axiom schema $T_{4}$, of requiring that $n$ connected sets of pairwise distance greater than $2^{n}$ can be colored independently, and the associated Claims 2.1 and 2.2, their proofs, and their application in proving claim (***), are formally similar to those of Section 3 of Chernikov, Hrushovski, Kruckman, Krupiński, Moconja, Pillay and Ramsey ([4]).

Let $\mathcal{L}$ be the language with sorts $P$ and $O$, symbols $R_{1}$ and $R_{2}$ for binary relations on $O$, and symbols $\rho_{1}$ and $\rho_{2}$ for binary relations between $P$ and $O$. Call an $\mathcal{L}$-structure $A$ copacetic if:

(C1) For $i=1,2$, $R_{i}(A)$ is a symmetric, irreflexive relation on $O(A)$, and the two are mutually exclusive: for $a_{1},a_{2}\in O(A)$, $A\not\models R_{1}(a_{1},a_{2})\wedge R_{2}(a_{1},a_{2})$.

(C2) The relation $R_{1}(A)\cup R_{2}(A)$ has no loops on $O(A)$ (i.e. there are no distinct $a_{0}\ldots a_{n-1}\in O(A)$, $n>2$, and $i_{1}\ldots i_{n}\in\{1,2\}$ so that, for $0\leq j\leq n-1$, $A\models R_{i_{j}}(a_{i},a_{i+1\mathrm{\>mod\>}n})$).

(C3) For all $b\in P(A)$, $a\in O(A)$, exactly one of $A\models\rho_{1}(b,a)$ and $A\models\rho_{2}(b,a)$ hold.

(C4):

(a) For each $b\in P(A)$, there are no distinct $a_{1},a_{2}\in O(A)$, so that there there is some $a_{*}\in O(A)$ so that $A\models R_{1}(a_{1},a_{*})\wedge R_{1}(a_{2},a_{*})$, and $A\models\rho_{1}(b,a_{1})\wedge\rho_{1}(b,a_{2})$.

(b) For each $b\in P(A)$, there are no distinct $a_{1},a_{2},a_{3}\in O(A)$, so that there there is some $a_{*}\in O(A)$ so that $A\models R_{2}(a_{1},a_{*})\wedge R_{2}(a_{2},a_{*})\wedge R_{2}(a_{3},a_{*})$, and $A\models\rho_{2}(b,a_{1})\wedge\rho_{2}(b,a_{2})\wedge\rho_{2}(b,a_{3})$.

Let $A$ be copacetic, and let $b\in P(A)$, $a\in O(A)$, $i\in{1,2}$. Then there are at most $i$ many $a^{\prime}\in O(A)$ with $A\models R_{i}(a,a^{\prime})$ and $A\models\rho_{i}(b,a^{\prime})$. For $N$ the number of such $a^{\prime}$ that are defined, let $b^{A,j}_{\to^{i}}(a)$, $1\leq j\leq N$, denote the $N$ many such $a^{\prime}$ (so if $i=2$ and there are two such $a^{\prime}$, make an arbitrary choice of which is $b^{A,1}_{\to^{i}}(a)$ and which is $b^{A,2}_{\to^{i}}(a)$, while if $i=1$, then $b^{A,1}_{\to^{i}}(a)$ is the sole such $a^{\prime}$, if it exists.)

If $B$ is copacetic and $A\subset B$ is a substructure of $B$ (and is therefore also copacetic), call $A$ closed in $B$ (denoted $A\leq B$) if

(i) For $b\in P(A)$, $a\in O(A)$, $i\in\{1,2\}$, $1\leq j\leq i$, if $b^{B,j}_{\to^{i}}(a)$ exists, then $b^{B,j}_{\to^{i}}(a)\in A$.

(ii) Any $R_{1}(B)\cup R_{2}(B)$-path between nodes of $O(A)$ lies in $O(A)$: for all $a_{1},a_{n}\in O(A)$, $a_{2},\ldots a_{n-1}\in O(B)$ which are distinct and distinct from $a_{1}$, $a_{n}$, and $i_{1},\ldots,i_{n-1}\in\{1,2\}$, if $A\models R_{i_{j}}(a_{i},a_{i+1})$ for $1\leq j\leq n-1$, then for all $2\leq i\leq n-1$, $a_{i}\in O(A)$.

Call a copacetic $\mathcal{L}$-structure $A$ connected if $R_{1}(A)\cup R_{2}(A)$ forms a connected graph on $O(A)$: for all $a,a^{\prime}\in O(A)$, there are $a_{2},\ldots a_{n-1}\in O(A)$, $i_{1},\ldots,i_{n-1}\in\{1,2\}$ for some $n$, so that for $a_{1}=a$, $a_{n}=a^{\prime}$, $A\models R_{i_{j}}(a_{i},a_{i+1})$ for all $1\leq j\leq n-1$. So if $B$ is copacetic and $A\subseteq B$, connectedness of $A$ supplants requirement (ii) of $A$ being closed in $B$. Call a subset of $A$ a connected component of $A$ if it is a maximal connected subset of $O(A)$.

Let $O$ be an undirected graph without cycles and with a $2$-coloring of its edges, with $R_{1}$, $R_{2}$ denoting edges of either color. Let $\rho_{1},\rho_{2}\subset O$, $O=\rho_{1}\cup\rho_{2}$, $\rho_{1}\cap\rho_{2}=\emptyset$ be a coloring of the vertices of $O$ so that, for $i=\{1,2\}$, no $i+1$ distinct vertices of $O$, lying on the boundary of the same $R_{i}$-ball of radius $1$ (i.e. they have a common $R_{i}$-neighbor), are both colored by $\rho_{i}$. Then we call $\rho_{1},\rho_{2}$ a (C4)-coloring of $O$. For $A$ copacetic, $O^{\prime}\subseteq O(A)$, and $\rho_{1},\rho_{2}\subseteq O^{\prime}$ a (C4)-coloring of $O^{\prime}$, say that $b\in P(A)$ induces the (C4)-coloring $\rho_{1},\rho_{2}$ on $O^{\prime}$ if for $i\in\{1,2\}$, $\rho_{i}=\{a\in O^{\prime}:A\models\rho_{i}(b,a)\}$.

The following assumptions on an $\mathcal{L}$-structure $A$ are expressible by a set of first-order sentences:

($T_{1}$) (Copaceticity) $A$ is copacetic.

($T_{2}$) (Completeness) For $b\in P(A)$, $a\in O(A)$, $i\in\{1,2\}$, if $A\models\neg\rho_{i}(b,a)$, then $b^{A,j}_{\to^{i}}(a)$ exists for $1\leq j\leq i$.

($T_{3}$) (Tree extension) For $C\subseteq A$, and $B\geq C$ finite and copacetic with $P(B)=P(C)$, there is an embedding $\iota:B\hookrightarrow A$ with $\iota|_{C}=\mathrm{id}_{C}$.

($T_{4}$) (Parameter introduction) For any $n<\omega$, finite connected sets $O_{1},\ldots,O_{n}\subseteq O(A)$ so that there does not exist an $R_{1}\vee R_{2}$-path of length at most $2^{n}$ between a vertex of $O_{i}$ and a vertex of $O_{j}$ for any distinct $i,j\leq n$ (so in particular, $O_{i}$ and $O_{j}$ are disjoint), and (C4)-colorings $\rho^{i}_{1},\rho^{i}_{2}\subseteq O_{i}$ of $O_{i}$ for $i\leq n$, there are infinitely many $b\in P(A)$ so that, for each $i\leq n$, $b$ induces the (C4)-coloring $\rho^{i}_{1},\rho^{i}_{2}\subseteq O_{i}$ on $O_{i}$.

Let $T^{\not\exists}=T_{1}\cup T_{2}\cup T_{3}\cup T_{4}$. We claim that $T^{\not\exists}$ is consistent. This will follow by induction from the following three claims:

(*) If $A$ is copacetic, $b\in P(A)$, $a\in O(A)$, $i\in\{1,2\}$, and $1\leq j\leq i$, there is a copacetic $\mathcal{L}$-structure $A^{\prime}\supseteq A$ (i.e. containing $A$ as a substructure) so that $b^{A^{\prime},j}_{\to^{i}}(a)$ exists.

(**) If $A$ is copacetic, $C\subseteq A$, and $B\geq C$ is finite and copacetic with $P(B)=P(C)$, there is a copacetic structure $A^{\prime}\supseteq A$ and an embedding $\iota:B\hookrightarrow A^{\prime}$ with $\iota|_{C}=\mathrm{id}_{C}$.

(***) If $A$ is copacetic, for any $n<\omega$, finite connected sets $O_{1},\ldots,O_{n}\subseteq O(A)$ so that there does not exist an $R_{1}\vee R_{2}$-path of length at most $2^{n}$ between a vertex of $O_{i}$ and a vertex of $O_{j}$ for any distinct $i,j\leq n$, and (C4)-colorings $\rho^{i}_{1},\rho^{i}_{2}\subseteq O_{i}$ of $O_{i}$ for $i\leq n$, there is some copacetic $A^{\prime}\supset A$ and $p\in P(A^{\prime}\backslash A)$ so that, for each $i\leq n$, $p$ induces the (C4)-coloring $\rho^{i}_{1},\rho^{i}_{2}$ on $O_{i}$.

We first show (*). Suppose $b^{A,j}_{\to^{i}}(a)$ does not already exist. Let $A^{\prime}=A\cup\{*\}$ for $*$ a new point of sort $O$, and let $R_{i}(A^{\prime})=R(A)\cup\{(a,*),(*,a)\}$, $R_{3-i}(A^{\prime})=R_{3-i}(A)$, $\rho_{3-i}(A^{\prime})=\rho_{3-i}(A)\cup(P(A)\backslash\{b\})\times\{*\}$, $\rho_{i}(A^{\prime})=\rho_{i}(A)\cup\{(b,*)\}$. Then $A^{\prime}$ is copacetic; first, (C1)-(C3) are clearly satisfied. Second, no point of $P(A)\backslash\{b\}$ can witness a failure of (C4), because $A$ is copacetic and $*$ is not an $R_{3-i}$-neighbor of any point of $O(A^{\prime})$. Finally, neither can $b$ witness a failure of (C4), because $*$ is not an $R_{i-3}$ neighbor of any point of $O(A^{\prime})$ and $a$ is the unique $R_{i}$-neighbor of $*$ in $O(A^{\prime})$, that failure must by copaceticity of $A$ be witnessed on the boundary of the $R_{i}$-ball of radius $1$ centered at $a$. But because $b^{A,j}_{\to^{i}}(a)$ does not already exist, there are fewer than $i$ many $R_{i}$-neighbors $a^{\prime}$ of $a$ in $O(A)$ with $A\models\rho_{i}(b,a^{\prime})$, so there are at most $i$ many $R_{i}$-neighbors $a^{\prime}$ of $A$ in $O(A^{\prime})$ with $A\models\rho_{i}(b,a^{\prime})$; thus the failure of (C4) is not in fact witnessed on this ball’s boundary. By construction, we can then choose $b^{A^{\prime},j}_{\to^{i}}(a)=*$.

We next show (**). For this we need the following claim:

Now, take the set $A^{\prime}$ to be the disjoint union of $A$ and $B$ over $C$; for notational simplicity we identify $A$, $B$ and $C$ respectively with their images in $A^{\prime}$. Let the $R_{1},R_{2}$-structure on $O(A^{\prime})$ be given as follows: the identification on $O(A)$ and $O(B)$ preserves the $R_{i}$-structure, and $O(A)$ and $O(B)$ are freely amalgamated over $O(C)$ (i.e. there are no $R_{i}$-edges between $O(A\backslash C)$ and $O(B\backslash C)$). This guarantees (C1), and also guarantees (C2) by condition (ii) of $C\leq B$. Now let the $\rho_{1},\rho_{2}$-structure on $A^{\prime}$ be given as follows: the identifications on $A$ and $B$ preserve the $\rho_{i}$-structure, which, by the assumption that $P(B)=P(C)$, leaves us only by way of satisfying (C3) to define the $\rho_{1},\rho_{2}$-structure on $P(A\backslash C)\times O(B\backslash C)$, and this will be the following. Let $p\in P(A\backslash C)$; the requirement that the $\rho_{i}$-structure on $A$ is preserved tells us the (C4)-coloring that $p$ induces on $O(C)$, and we extend this to a (C4)-coloring on $O(B)$ as in Claim 2.1; here we use condition (ii) of $C\leq B$. We have defined the full $\mathcal{L}$-structure on $A^{\prime}$, so it remains to show (C4); in other words, we show that $p$ induces a (C4)-coloring on $A^{\prime}$ in the case where $p\in P(C)$ and in the case where $p\in P(A\backslash C)$. In either case $p$ induces a (C4)-coloring on $A$ and on $B$, so the only way (C4) can fail is on the boundary of an $R_{i}$-ball of radius $1$ centered at $c\in C$. But (C4) cannot fail on the boundary of this ball, because it does not fail there in $A$, and by condition (i) of $C\leq B$ in the first case, or by the additional clause of Claim 2.1 in the second case, any $R_{i}$-neighbor of $c$ in $B\backslash C$ is colored by $\rho_{3-i}$ in the (C4)-coloring induced by $p$ on $B$, so (C4) still cannot fail on the boundary of this ball in $A^{\prime}$. So $A^{\prime}$ is a copacetic $\mathcal{L}$-structure containing $A$, and clearly, the embedding $\iota$ as in (**) exists, so this shows (**).

Finally, we show (***). For this, we need an additional combinatorial claim:

By Claims 2.2 and 2.1, there is a (C4)-coloring $\rho_{1},\rho_{2}$ on $O(A)$ extending the (C4)-coloring $\rho^{i}_{1},\rho^{i}_{2}$ on $O_{i}$ for each $i\leq n$. So we can extend $A$ to $A^{\prime}=A\cup p$, where $p\in P(A^{\prime}\backslash A)$, and define the relations $\rho_{1}$ and $\rho_{2}$ on $O(A)\times\{p\}$ so that $p$ induces the (C4)-coloring $\rho_{1},\rho_{2}$ on $O(A)$. This proves (***), and the consistency of $T^{\not\exists}$.

We call a copacetic $\mathcal{L}$-structure complete if it satisfies axiom $T_{2}$; completeness of $A$ will always supplant condition (i) of $A\leq B$.

We now prove that $T^{\not\exists}$ has the following embedding property:

We now fix a sufficiently saturated $\mathbb{M}\models T^{\not\exists}$, which we take as the ambient model; for $p\in P(\mathbb{M})$, fix the notation, $p^{j}_{\to^{i}}=:p^{\mathbb{M},j}_{\to^{i}}$ The following quantifier elimination is a corollary of the above:

Next, we show that $T^{\not\exists}$ is $\mathrm{NSOP}_{1}$. By Theorem 9.1 of [12], it suffices to find an invariant ternary relation $\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}$ between subsets of $\mathbb{M}$ over models with the following properties:

(a) Strong finite character: For all $a,b$ and $M\models T$, if $a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}b$, then there is some formula $\varphi(x,b)\in\mathrm{tp}(a/Mb)$, where $\varphi(x,y)$ has parameters in $M$, such that for every $a^{\prime}\models\varphi(x,b)$, $a^{\prime}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}b$.

(b) Existence over models: for all $a$ and $M\models T$, $a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}M$.

(c) Monotonicity: For all $A^{\prime}\subseteq A$, $B^{\prime}\subseteq B$, $M\models T$, $A\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}B$ implies $A^{\prime}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}B^{\prime}$.

(d) Symmetry: For all $a,b$ and $M\models T$, $a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}b$ implies $b\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}a$ (and vice versa).

(e) The independence theorem: for any $a,a^{\prime},b,c$ and $M\models T$, $a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}b$, $a^{\prime}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}c$, $b\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}c$ and $a\equiv_{M}a^{\prime}$ implies that there is $a^{\prime\prime}$ with $a^{\prime\prime}b\equiv_{M}ab$, $a^{\prime\prime}c\equiv_{M}a^{\prime}c$ and $a^{\prime\prime}\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}bc$.

First, note that for any $A$, there is $A^{\prime}\supseteq A$ such that $A^{\prime}\leq\mathbb{M}$ and for all $A^{\prime\prime}\supseteq A$ with $A^{\prime\prime}\leq\mathbb{M}$, $A^{\prime}\subseteq A^{\prime\prime}$. Denote this $\mathrm{cl}(A)$; clearly, this is contained in $\mathrm{acl}(A)$ (and it can even be checked that $\mathrm{cl}(A)=\mathrm{acl}(A)$). This allows us to define $\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}$: $A\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}B$ if $\mathrm{cl}(MA)\cap\mathrm{cl}(MB)=M$, and every ($R_{1}(\mathbb{M})\cup R_{2}(\mathbb{M})$)-path between a point of $O(\mathrm{cl}(MA)\backslash M)$ and $O(\mathrm{cl}(MB)\backslash M)$ contains a point of $M$.

We see this has strong finite character: for $a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mathchar 12854\relax$\kern 8.00134pt\hss}\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}b$, $a_{0}\in\mathrm{cl}(aM)$, $b_{0}\in\mathrm{cl}(bM)$ the endpoints of the path of length $n$ witnessing this, let $\varphi_{1}(y,b)$ isolate $b_{0}$ over $Mb$, $\varphi_{2}(x)$ say that $x$ is not within distance $n$ of the closest point of $M$ to some (any) $b^{\prime}_{0}\models\varphi_{1}(x,b)$ (which would be required, should there be a (non-self-overlapping) path of length $n$ between $x$ and $b_{0}$ going through $M$), and $\varphi_{3}(x,b)$ say that there is a path of length $n$ from $x$ to some point $c$ satisfying $\models\varphi_{1}(c,b)$. Then $\varphi(x,b)=:\varphi_{2}(x)\wedge\varphi_{3}(x,b)$ suffices.

Existence over models, monotonicity, and symmetry are immediate. So it remains to show the independence theorem. By definition of $\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}$, we may assume $a=\mathrm{cl}(aM)$ and similar for $a^{\prime},b,c$. We first give an analysis of the structure of pairs $a,b$ with $a\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\displaystyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\textstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptstyle{}}{\kern 5.71527pt\hbox to0.0pt{\hss$\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\smile$\hss}\kern 5.71527pt\scriptscriptstyle{}}}_{M}b$: