Generic derivations on algebraically bounded structures II
Model theoretical properties
July 29, 2025
Let T be an algebraically bounded theory. We consider the \(L(\bar\delta)\)-expansions of T by a tuple \(\bar \delta\) of derivations (which may be commuting or not). We investigate the model completion of either of the above theories, whose existence has been established in [@FT:24], with particular attention to its model-theoretic properties, including \(\omega\)-stability, simplicity, open core, and elimination of imaginaries.
We study the model-theoretic properties of the model completion of the theory of algebraically bounded fields expanded with derivations, continuing the foundational work in [@FT:24]. Algebraically bounded fields, encompassing structures like algebraically closed, real closed, and p-adically closed fields, provide a rich framework for exploring the interplay between field-theoretic and model-theoretic algebraic notions. Throughout this article, we denote by \(\mathbb{K}\) a structure that expands a field of characteristic 0. Recall that \(\mathbb{K}\) is algebraically bounded if the model-theoretic algebraic closure and the field-theoretic algebraic closure coincide in every structure elementarily equivalent to \(\mathbb{K}\); for more details, examples, and main properties see [@Dries:89].
Let \(\mathbb{K}\) be algebraically bounded, \(L\) be the language of \(\mathbb{K}\), and \(T\) be its theory. In order to study derivations on \(\mathbb{K}\), we denote by \(\bar\delta= \{\delta_1, \ldots, \delta_k\}\) new unary functions symbols.
Assume that \(T\) is model complete. In [@FT:24] we proved that, for every \(k \in \mathbb{N}\), the \(L^{\bar
\delta}\)-expansions of \(T\), \(T^{\bar \delta,nc}\) (saying that \(\bar\delta\) is a \(k\)-tuple of derivations)
and \(T^{\bar \delta}\) (saying that \(\bar\delta\) is a \(k\)-tuple of commuting derivations) have a model completion. We denote by \(T^{\bar \delta,?}_g\) the model completion of either of the above theories.
In the preceding work [@FT:24], we began investigating the model-theoretic properties of \(T^{\bar \delta,?}_g\): we showed that if \(T\) is
stable (resp., dependent), then \(T^{\bar \delta,?}_g\) is also stable (resp., dependent).
The above results include and extend a long series of results in the literature (see [@FT:24] for a short survey: here we mention [@Tressl:05] that shows that large model complete fields admit generic derivations). There are also examples of theories which do not admit generic derivations (see [@FT:dexp]).
In this paper, we extend the investigation of \(T^{\bar \delta,?}_g\), with a specific focus on model-theoretic properties such as stability, \(\omega\)-stability, simplicity, and elimination of imaginaries; we also address other questions raised in [@FT:24]. Multiple authors have examined these properties in several particular cases; here we consider general setting of algebraically bounded fields with several generic derivations. For instance, in [@MS], the authors demonstrated that \(\mathop{\mathrm{DCF}}_{0, m, nc}\), the model completion of the theory of fields with non-commuting derivations, is stable but not \(\omega\)-stable, eliminates imaginaries, and is uniformly finite. Building upon these results, [@Mohamed; @SanchM:24] further generalized some of the properties established in [@MS].
In this article, we pay particular attention to the following questions:
When is \(T^{\bar \delta,?}_g\) simple or totally transcendental?
Which are the imaginaries of \(T^{\bar \delta,?}_g\)?
If \(T\) is equipped with a definable topology, is \(T\) the open core of \(T^{\bar \delta,?}_g\)?
Is \(T^{\bar \delta,?}_g\) Uniformly Finite?
What is a “generic” object?
Does \(T^{\bar \delta,?}_g\) have a dimension function in the sense of [@Dries:89]?
What is the theory of the restriction of \(T^{\bar \delta,?}_g\) where we forget the derivation but we keep the field of constants?
We will give a full answer to 1) ([sec:simple] [sec:w-stable] and [@FKM-26]). In 7 we prove that if \(T\) is simple, then \(T^{\bar \delta,?}_g\) is simple. [@MS] showed that the model completion of the theory of fields with non-commuting operators is simple, eliminates imaginaries, is uniformly finite, and under some additional assumptions it is stable.
In [@FKM-26] the first author together with E. Kaplan and A. Matthews further extend the results exposed here, proving the following:
Theorem A 1.
\(T\) is rosy iff it is superrosy of thorn rank 1; in this case, \(T\) has Geometric Elimination of Imaginaries;
\(T\) is simple iff it is supersimple or SU-rank 1;
\(T\) is stable iff it is equal to the theory of pure algebraically closed fields, expanded by some constants (we will simply write “\(T = \mathop{\mathrm{ACF_0}}\)”).
Combining the above with the results in this paper, we get that \(T^{\bar \delta,?}_g\) is totally transcendental iff \(T = \mathop{\mathrm{ACF_0}}\) iff \(T\) is stable.
We have some partial answers to 2). We show that if \(T\) has (Geometric) Elimination of Imaginaries and it is either stable or supersimple, then \(T^{\bar \delta,?}_g\) also has (Geometric) EI: 31 34 4: together with Theorem A, we get that if \(T\) is simple then \(T^{\bar \delta,?}_g\) has GEI. We also show that if \(T\) has a definable topology satisfying some general conditions, then \(T^{\bar \delta,?}_g\) eliminates imaginaries relative to \(T\): see 5; we generalize results in [@KP]. We conjecture that a stronger result holds than what we manage to prove (see 8 26).
We give a complete answer to 3) (11 in particular 5).
For 4) we will give a positive answer in a separate paper.
For 5) we give some possible answers (13), where we study the Polish space of all derivations on a countable model \(M \models T\) (under some “bigness” condition on \(M\)) and show that generic derivations form a dense \(\mathcal{G}_{\delta}\)-set among all possible derivations.
For 6) we give a complete answer in 12: \(T^{\bar \delta,?}_g\) has a dimension function iff the derivations commute. We also generalize the result in [@ELR] on “coincidence of dimensions”: see also [@GP:12; @BMR] for previous results in particular cases.
For 7), in 10 we study the field of constant \(\mathcal{C}_{\bar \delta}\) of a model \(\langle \mathbb{K}, \bar \delta \rangle \models T^{\bar \delta,?}_g\). We show that \(\langle \mathbb{K}, \mathcal{C}_{\bar \delta} \rangle\) is a lovely pair of geometric structures (in the sense of [@BerensteinV:10]), and we study the definable subsets of \(\mathcal{C}_{\bar \delta}^{n}\).
In [sec:ext] [sec:ind] we study independence relations on model of \(T\) and of \(T^{\bar \delta,?}_g\). We introduce the independence relation \(\makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}\) and prove the fundamental Extension Theorem 21 and Independence Theorem 22: they will make much easier to prove the results in [sec:simple] [sec:open]. We also characterize the algebraic closure inside models of \(T^{\bar \delta,?}_g\).
Let us mention here the recent work [@PillayPR:25] where the authors study groups definable in models of \(T^{\bar \delta}_g\): they show that any such group can be embedded in a \(T\)-definable group.
We conclude the paper with several open questions, conjectures, and announcements of further work.
\(L\) is a language and \(T\) is a \(L\)-theory (later we will impose additional conditions on them).
When we say that \(\mathfrak C\) is a “monster model”, we mean that \(\mathfrak C\) is a \(\lambda\)-saturated and \(\lambda\)-homogeneous model of \(T\) for a “sufficently big” cardinal \(\lambda > \aleph_{0} + \lvert L\rvert\) (other authors mean that \(\mathfrak C\) is a proper class, and it is \(\lambda\)-saturated and \(\lambda\)-homogeneous for every cardinal \(\lambda\)).
By “small” subset/tuple of \(\mathfrak C\) we mean of cardinality less than \(\lambda\).
\(\mathbb{K}\) will be an \(L\)-structure, and \(K\) will be either its domain, or, when \(\mathbb{K}\) expands a field, its underlying field.
We say that \(\mathbb{K}\) is algebraically bounded if it expands a field of characteristic 0 and, for every subset \(A \subseteq K\) (not necessarily a substructure), the model theoretic closure of \(A\) (computed in a sufficiently saturated elementary extension of \(\mathbb{K}\)) coincides with the field theoretic algebraic closure of the field generated by \(A\) and \(\mathrm{dcl}(\emptyset)\).
This work explores the model theory of differential fields, specifically focusing on expansions of a base theory \(T\) by derivations.
Our assumptions for the whole article are the following:
\(\mathbb{K}\) is a structure expanding a field of characteristic \(0\).
\(L\) is the language of \(\mathbb{K}\) and \(T = Th(\mathbb{K})\) is its model complete \(L\)-theory.
\(F \mathrel{\vcenter{:}}= \mathop{\mathrm{dcl}}(\emptyset) \subseteq \mathbb{K}\).
\(\mathbb{K}\) is algebraically bounded over \(F\).
By algebraic closure we always mean the \(T\)-algebraic closure; in particular, \(\mathop{\mathrm{acl}}\) denotes the \(T\)-algebraic closure, and algebraic independence is understood with respect to \(T\) (equivalently, over \(F\) in the field-theoretic sense).
Under our assumptions, \(\mathbb{K}\) is geometric: in the monster model \(\mathfrak C\succ \mathbb{K}\), the operator \(\mathop{\mathrm{acl}}\) satisfies the exchange property, and thus induces a matroid structure.
Moreover, \(\mathbb{K}\) is equipped with a dimension function \(\dim\), assigning to each definable set (with parameters) a natural number and satisfying the axioms of [@Dries:89].
We also consider the rank function \(\mathop{\mathrm{rk}}\) associated with the matroid \(\mathop{\mathrm{acl}}\): for sets \(V\) and \(B\), \(\mathop{\mathrm{rk}}(V/B)\) is the cardinality of a basis of \(V\) over \(B\). In particular, if \(X \subseteq M^n\) is definable with parameters \(\bar b\), then \[\dim(X) = \max \{ \mathop{\mathrm{rk}}(\bar a / \bar b) : \bar a \in X \}.\]
Let \(\bar \delta:= \langle \delta_{1}, \dotsc, \delta_{k} \rangle\). Let \(\eta_{1}, \dotsc, \eta_{k}\) be derivations on \(F\). We obtain two theories:
the expansion of \(T\) saying that the \(\delta_{i}\) are derivations which commute with each other, i.e. for every \(i\), \(j \leq k\), \(\delta_{i} \circ \delta_{j} = \delta_{j} \circ \delta_{i}\), and that \(\delta_{i}\) extends \(\eta_{i}\) for \(i \leq k\);
the expansion of \(T\) saying that the \(\delta_{i}\) are derivations without any further conditions and that \(\delta_{i}\) extends \(\eta_{i}\) for \(i \leq k\).
Both theories have a model completion (remember that \(T\) is model complete) (see [@FT:24]). For convenience, we use \(T^{\bar
\delta,?}_g\) to denote either of the model completions, both for commuting and non-commuting tuples of derivations.
\(\langle \mathfrak C, \bar\delta \rangle\) is a monster model of \(T^{\bar \delta,?}_g\).
\(\langle \mathbb{K}, \bar\delta \rangle\) is some model of \(T^{\bar \delta,?}_g.\)
We introduce some notation that we use in the sequel.
We denote by \(\Gamma\) the free commutative (or non commutative) monoid generated by \(\bar \delta,\) we can define the canonical partial order \(\preceq\)
given by \(\beta \preceq \alpha\beta\), for all \(\alpha, \beta \in \Gamma\).
Remark 1. If \(\Gamma\) is the free non commutative monoid generated by \(\bar \delta\) then \(\preceq\) is a well-founded partial order on \(\Gamma\), but it is not a well-partial-order (i.e., there exist infinite anti-chains).
\(\emptyset\) (i.e., the empty word, corresponding to the identity function on \(\mathbb{K}\)) is the minimum of \(\Gamma\);
If \(\alpha \preceq \beta\), then \(\gamma \alpha \leq \gamma \beta\) and \(\alpha \gamma \leq \beta \gamma\).
Remark 2. If \(\Gamma\) is the free commutative monoid generated by \(\bar \delta\), with the canonical partial order \(\preceq\), it is isomorphic to \(\mathbb{N}^k\).
Fix \(n \in \mathbb{N}\), \(\bar x= \langle x_{1}, \dotsc, x_{n} \rangle\), and denote by \(\Gamma_{n} \mathrel{\vcenter{:}}= \{\gamma x_{i}: \gamma \in \Gamma, i \leq n\}\): we consider \(\Gamma_{n}\) as a set of some \(L^{\bar \delta}\)-terms, as a set of functions from \(K^{n}\) to \(K\), and as a set of indices. For example \(\delta_1\delta_2x_3\) is in \(\Gamma_{3}\) but \(\delta_1\delta_2x_3 + \delta_1x_1\) is not. Given \(\bar a \in \mathbb{K}^n\), we denote by \(\mathop{\mathrm{Jet}}(\bar a) := \langle \gamma \bar a: \gamma \in \Gamma_n \rangle\).
Moreover, we denote by \(\bar x_{\Gamma}\) a set of formal variables indexed by \(\Gamma_{n}\): that is, for every \(f \in \Gamma_{n}\) we have a variable \(\bar x_{f}\). Given an \(L(A)\)-formula \(\alpha(\bar x_{\Gamma})\) we denote by \(\alpha(\Gamma)\) the \(L^{\bar \delta}(A)\)-formula where we replaces each occurrence of a variable \(\bar x_{f}\) with the corresponding term \(f(\bar x) \in \Gamma_{n}\).
Let \(J \subseteq \Gamma_{n}\) and \(A \subset K\) be a “small” subset of \(K\) with \(\bar \delta A \subseteq A\). Let \(p(\bar x)\) be a complete \(L^{\bar \delta}\)-type over \(A\) (in \(n\) variables). Define \[p_{J} \mathrel{\vcenter{:}}= \{ \alpha: \alpha(\bar x_\Gamma)\;L(A)\text{-formula s.t. } \alpha(\Gamma) \in p\} \in S_{L}^{J}(A).\] We say that some type \(q \in S^{J}_{L}(A)\) is \(\bar \delta\)-compatible if there exists \(s \in S^{n}_{L^{\bar \delta}}(A)\) such that \(q = s_{J}\).
Many of the model-theoretic properties of \(T\) are inherited by \(T^{\bar \delta,?}_g\). The following results from [@FT:24], which plays a crucial role in our subsequent analysis, is particularly useful:
Theorem 3. For every \(\bar a\) tuple in \(\mathfrak C\) and \(B\) subset of \(\mathfrak C\), the \(L^{\bar \delta}\)-type of \(\bar a\) over \(B\) is uniquely determined by the \(L\)-tuple of \(\mathop{\mathrm{Jet}}(\bar a)\) over \(\mathop{\mathrm{Jet}}(B)\).
Theorem 4.
If \(T\) eliminates quantifiers, then \(T^{\bar \delta,?}_g\) eliminates quantifiers.
For every \(L^{\bar \delta}\)-formula \(\alpha(\bar x)\) there exists an \(L\)-formula \(\beta(\bar x_\Gamma)\) such that \[T^{\bar \delta,?}_g\models \forall \bar x\;\bigl( \alpha(\bar x) \leftrightarrow \beta(\Gamma) \bigr).\]
Elimination of imaginaries is a powerful concept in model theory that simplifies arguments by allowing us to work directly with elements of the model, rather than with equivalence classes of definable sets.
We remind what Elimination of Imaginaries and some of its variants are.
Definition 1. \(T\) eliminates imaginaries (\(T\) has EI) if every imaginary is interdefinable with a real tuple: i.e., for every imaginary \(e\) there is a real tuple \(a\) such that \(\mathop{\mathrm{dcl^{eq}}}(e) = \mathop{\mathrm{dcl^{eq}}}(a).\)
Definition 2. \(T\) weakly eliminates imaginaries (\(T\) has WEI), if for every imaginary \(e\) there is a real tuple \(a\) such that \(e\) is definable over \(a\) and a is algebraic over \(e\): that is, \(e \in \mathop{\mathrm{dcl^{eq}}}(a)\) and \(a \in \mathop{\mathrm{acl^{eq}}}(e).\)
Remark 5. Equivalently, a theory \(T\) admits weak elimination of imaginaries (WEI) in the sense of B. Poizat (see [@CasanF:04; @Yoneda:22]) if, for any \(\phi(\overline{x}, \overline{a}) \in L(\overline{a})\), we have the smallest algebraically closed set \(B\) such that \(\phi(\overline{x}, \overline{a})\) is definable over \(B\).
Definition 3. \(T\) has geometric eliminates imaginaries (\(T\) has GEI), if for every imaginary \(e\) there is a real tuple \(a\) such that \(e\) is algebraic over \(a\) and \(a\) is algebraic over \(e\): that is, \(e \in \mathop{\mathrm{acl^{eq}}}(a)\) and \(a \in \mathop{\mathrm{acl^{eq}}}(e)\).
Remark 6. By definition, it is clear that EI implies WEI, and that WEI implies GEI. The converse implication, namely that WEI implies EI, holds when working over a field as is the case here (see [@MMP]).
Remark 7. In any structure containing at least two constants, as in our case, the definition of EI is equivalent to the Uniform Elimination of Imaginaries (see [@CasanF:04] for definition and proof).
Regarding generic derivations, we formulated the following:
Conjecture 8 ([@FT:24]). Let \(T\) be algebraically bounded. Then, \(T^{\bar \delta,?}_g\) has elimination of imaginaries modulo \(T^{eq}\).
In particular: if \(T\) has GEI, then \(T^{\bar \delta,?}_g\) also has GEI, and if \(T\) has EI, then \(T^{\bar \delta,?}_g\) also has EI.
A few particular cases were already known, when \(T\) is one of the following:
\(\mathop{\mathrm{ACF_0}}\): see [@mcgrail; @MS];
\(\mathop{\mathrm{RCF}}\): see [@FK; @KP] for a proof based on M. Tressl’s idea, see also [@bkp; @Point] for different proofs.
We will prove two cases of the above conjecture:
When \(T\) is simple: by Theorem A \(T\) has GEI; we then employ a result in [@Yoneda:09] to show that \(T^{\bar \delta,?}_g\) also has GEI(see 7.1); if \(T\) has EI, we adapt a technique in [@HC:99] to show that \(T^{\bar \delta,?}_g\) also has EI (see 7.2).
When \(T\) has a suitable definable topology: we employ a technique by M. Tressl to show that \(T^{\bar \delta,?}_g\) has elimination of imaginaries modulo \(T^{eq}\) (see 11).
To establish the primary results of this paper, we remind the concept of independence relations and revisit key theorems related to this notion. We fix a monster model \(\mathfrak C\models T\).
Let \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) be a ternary relation on small subsets of \(\mathfrak C\). We mostly use the definitions and nomenclature from [@Adler]: in particular, we are interested in the cases when \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is either a strict independence relation, or the relation \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}\) defined in [@Adler], or Shelah-forking \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\).
We remind for completeness the definition \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}\) introduced in [@Adler].
Definition 4. The relation \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}\) (M-dividing independence) is defined as:
\(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}_C B\) iff for any \(C'\) s.t \(C \subseteq C' \subseteq \mathop{\mathrm{acl}}(BC)\) then \(\mathop{\mathrm{acl}}(AC') \cap \mathop{\mathrm{acl}}(BC') = \mathop{\mathrm{acl}}C'\).
For the remainder of this section, \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is an independence relation on \(\mathfrak C\).
Remark 9. If \(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C} B\), then \(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C} \mathop{\mathrm{acl}}(BC)\).
Proof. By Existence, there exists \(D \equiv_{BC}{\mathop{\mathrm{acl}}(BC)}\) s.t. \(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{BC} D\): notice that \(D = \mathop{\mathrm{acl}}(BC)\), and therefore \(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{BC} \mathop{\mathrm{acl}}(BC)\). By transitivity, \(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C} \mathop{\mathrm{acl}}(BC)\). ◻
Notice that we don’t assume that \(T\) has some form of elimination of imaginaries, nor that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) extends to \(\mathfrak C^{eq}\). However, the setting in [@Adler] often assumes implicitly to work in \(T^{eq}\): to avoid confusion, we will introduce some additional nomenclature, following in part [@Yoneda:09].
Definition 5. We say that an independence relation \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is:
if, for every \(a \in \mathfrak C\) and \(B \subset \mathfrak C\), \(a \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{B} a\) iff \(a \in \mathop{\mathrm{acl}}(B)\);
if \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) extends to an independence relation on \(\mathfrak C^{eq}\) (which we also denote by \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\)), and for every \(a \in \mathfrak C^{eq}\) and \(B \subset \mathfrak C^{eq}\), \(a \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{B} a\) iff \(a \in \mathop{\mathrm{acl^{eq}}}(B)\);
if it is real-strict and, for every \(A, B, C, D \subset \mathfrak C\) such that \(B \supseteqq C, D\) \[A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C} B \ \wedge\ A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{D} B \implies A\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{\mathop{\mathrm{acl}}(C) \cap \mathop{\mathrm{acl}}(D)} B;\]
if it is strict and, for every \(A, B, C, D \subset \mathfrak C^{eq}\), such that \(B \supseteqq C, D\) \[A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C} B \ \wedge\ A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{D} B \implies A\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{\mathop{\mathrm{acl^{eq}}}(C) \cap \mathop{\mathrm{acl^{eq}}}(D)} B\]
Notice that if \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is strict, then it is real-strict. If either \(\mathfrak C= \mathfrak C^{eq}\), or \(\mathfrak C\) has GEI(Geometric Elimination of Imaginaries), then \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is real-strict iff it is strict; similarly, under the same assumption, \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is real-canonical iff it is canonical.
Fact 10.
There exist rosy theories with EI (even o-minimal) such that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}\) (thorn-forking independence) is not (real)-canonical: [@Adler] explains an example from [@LP:93]; see also [@Yoneda:09].
There exist a theory with WEI (the “integral Urysohn space”) and more than one strict independence relation: see [@Conant-16].
Fact 11 ([@Yoneda:09]). If \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is strict and real-canonical, then \(\mathfrak C\) has GEI.
Corollary 1. Let \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) be real-strict and real-canonical. T.f.a.e.:
\(\mathfrak C\) has GEI;
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is strict;
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}\).
Lemma 12. Let \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) be a real-strict independence relation on \(\mathfrak C\). If \(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C} B\), then \(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}_{C} B\).
In particular, if \(T\) is simple and \(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}_{C} B\), then \(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}_{C} B\).
Proof. Let \(A, B, C, C'\) be as in 4. Assume that \(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C} B\). Let \(d \in \mathop{\mathrm{acl}}(AC') \cap \mathop{\mathrm{acl}}(BC')\). We need to show that \(d \in \mathop{\mathrm{acl}}(C')\). \[\begin{gather} A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C} B \implies A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C} \mathop{\mathrm{acl}}(BC) \implies A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C} BC' \implies \\ \implies A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C'} B \implies \mathop{\mathrm{acl}}(AC') \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C'} \mathop{\mathrm{acl}}(BC') \implies d \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C'} d. \end{gather}\] Since \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is real-strict, the latter implies that \(d \in \mathop{\mathrm{acl}}(C')\). ◻
Fact 13 ([@BuechPW:00],[@Adler]).
If \(\mathfrak C\) is either stable or super-simple, then it has Elimination of HyperImaginaries (EHI);
If \(\mathfrak C\) is simple with EHI, then \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) is canonical;
If \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is canonical, then \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}\).
For more on EHI in (simple) theories, see e.g. [@Casanovas:11; @PalacW:13; @BuechPW:00]; there are no known examples of simple theories that do not eliminate hyperimaginaries; for us the relevant implication is the following fact:
Fact 14. If \(\mathfrak C\) is either stable or super-simple, then \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) is canonical and \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}\).
Definition 6. \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\)satisfies “Independence over Models” if:
For every \(\bar a, \bar b\in \mathfrak C^{n}\), \(\bar a', \bar b' \in \mathfrak C^{m}\), for every \(M \prec \mathfrak C\) with \(M\) small, if \[\bar a\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{M} \bar b, \qquad \bar a' \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{M} \bar a, \qquad \bar b' \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{M} \bar b, \qquad \bar a' \equiv^{L}_{M} \bar b'\] then there exists \(\bar c\in \mathfrak C^{m}\) such that \[\bar c\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{M} \bar a\bar b, \qquad \bar c\equiv^{L}_{M \bar a} \bar a', \qquad \bar c\equiv^{L}_{M \bar b} \bar b'.\]
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) satisfies “Independence over Algebraically Closed (real) Substructures” (IACS) if the above is true when we let \(M\) vary among algebraically closed substructure of \(\mathfrak C\). We say that \(T\) satisfies IACS if \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) satisfies it.
Fact 15 (Kim-Pillay: see [@KimP:98], [@TZ]). Let \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) be an independence relation on \(\mathfrak C\) that satisfies Independence over Models. Then, \(\mathfrak C\) is simple and \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is Shelah-forking (on \(\mathfrak C\)).
Conversely, if \(\mathfrak C\) is simple, then \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) is a strict independence relation satisfying Independence over Models.
Fact 16 ([@BuechPW:00]). If \(\mathfrak C\) is simple with EHI and EI, then \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) satisfies IACS.
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) can be extended to \(L\)-types: given \(A \subseteq B\) small subsets of \(\mathfrak C\) and an \(L(B)\)-type (in a small number of variables) \(q(\bar y)\), we write \[q \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{A} B\] if for some \(\bar c\in \mathfrak C^{\lvert\bar y\rvert}\) realizing \(q(\bar y)\) we have that \[\bar c\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{A} B.\] Notice that the above is equivalent to:
“For every \(\bar c\in \mathfrak C^{\lvert\bar y\rvert}\) realizing \(q(\bar y)\) we have that \(\bar c\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{A} B\)”.
We analyze now the case when \(L\) extends the language of rings and \(T\) is algebraically bounded.
Lemma 17. Let \(A, B, C \subseteq \mathfrak C\). T.f.a.e.:
\(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}_{C} B\);
for every \(A' \subseteq A\), if \(A'\) is algebraically independent over \(CF\) (where \(F = dcl(\emptyset)\)), then it is still algebraically independent over \(BC\);
for every \(A' \subseteq A\), if \(A'\) generates \(A\) (in the sense of the matroid \(\mathop{\mathrm{acl}}\)) over \(BC\), then it generates \(A\) over \(B\);
for every \(A' \subseteq A\) and \(B' \subseteq B\), \(\mathop{\mathrm{tr.deg.}}(A'B'/C) = \mathop{\mathrm{tr.deg.}}(A'/C) + \mathop{\mathrm{tr.deg.}}(B'/C)\).
Moreover, \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}\) is a real-strict independence relation on \(\mathfrak C\) satisfying the Strong Finite Character condition.
Proof. For the “moreover” part, notice that [en:ind-sum] implies that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}\) is symmetric, and the conclusion follows by [@Adler]. ◻
Fact 18. If \(\mathfrak C\) is algebraically bounded and has GEI, then \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}\) and \(\mathfrak C\) is rosy, i.e. \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}\) is an independence relation on \(\mathfrak C^{eq}\); moreover, it is super-rosy of -rank 1.1
Proof. Since \(\mathfrak C\) has GEI, \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}\) is a strict independence relation on \(\mathfrak C^{eq}\), and therefore \(\mathfrak C\) is rosy. For more details see [@Adler]: in particular, [@Adler].
\(\mathfrak C\) is then super-rosy of rank 1 because \(\mathop{\mathrm{U^{\text{\th}}}}(\bar a/B) =\mathop{\mathrm{tr.deg.}}(\bar a/ FB)\), where \(\mathop{\mathrm{U^{\text{\th}}}}\) is the -rank, \(\mathop{\mathrm{tr.deg.}}\) is the transcendence degree, and \(F\) is the model-theoretic algebraic closure of the empty set. ◻
Remark 19. Assume that \(\mathfrak C\) is algebraically bounded, with GEI, simple, and \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) is canonical. Then, \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\).
Proof. Since \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) is canonical, \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}\), and by the Fact 18 we have the conclusion. ◻
The results presented in this section are of independent interest and will be utilized in the sequel. For now, \(K\) is some field of characteristic \(0\).
Definition 7. We denote by \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}\) the ternary relation on subsets of \(K\) induced by \(\mathop{\mathrm{acl}}\) over \(F\): \[A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}_{C} B\] iff, for every \(\bar a\) finite subset of \(A\), if \(\bar a\) is algebraically independent over \(CF\), then \(\bar a\) remains algebraically independent over \(CFB\).2
The following lemma is well-known; we provide a proof for completeness.
Lemma 20 (Amalgamation). Let \(B_0\), \(B_1\), \(B_2\) be subrings of \(K\) containing \(F\). Let \(\bar \delta_i\) be \(k\)-tuples of derivations on \(B_i\), \(i=0, 1, 2\). Assume:
\(\langle B_0, \bar \delta_0 \rangle\) is a common substructure of \(\langle B_1, \bar \delta_1 \rangle\) and \(\langle B_2, \bar \delta_2 \rangle\);
\[B_{1} \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}_{B_{0}} B_{2}.\]
Then, there exists a \(k\)-tuple of derivations \(\bar \delta\) on \(K\) which extends all the \(\bar \delta_i\). Moreover, if each \(\bar \delta_i\) commutes, then \(\bar \delta\) commutes.
Proof. Let \(\bar b_i\) be a transcendence basis of \(B_i\) over \(B_0\), \(i = 1,2\). By 2), \(\bar b_1\) and \(\bar b_2\) are disjoint, and \(\bar b_1 \cup \bar b_2\) is algebraically independent over \(B_0\). Thus, there exists a \(k\)-tuple of derivations \(\bar \delta\) on \(K\) extending \(\bar \delta_{0}\) and such that \[\bar \delta(\bar b_i) = \bar \delta_i(\bar b_i), \qquad i= 1, 2.\] But then \(\bar \delta\) extends also \(\bar \delta_1\) and \(\bar \delta_2\). If moreover each \(\bar \delta_i\) commutes, let \(\bar c\) be a transcendence basis of \(K\) over \(B_0 \cup B_1 \cup B_2\). Let \(\bar \delta\) be the unique \(k\)-tuple of derivations on \(K\) such that:
\(\bar \delta\) extends \(\bar \delta_{0}\);
\(\bar \delta(\bar b_i) = \bar \delta_i(\bar b_i)\), \(i= 1, 2\);
\(\bar \delta(\bar c)\) = 0.
Then, \(\bar \delta\) commutes. ◻
Now \(\langle \mathbb{K}, \bar \delta \rangle\) is a monster model of \(T^{\bar \delta,?}_g\).
The following technical result has several applications: we will see some later; in [@PPP:23] the authors prove a version of it in order to study definable groups.
Theorem 21 (Extension). Let \(A \subseteq B\) be “small” subsets of \(K\) with \(\bar \delta A \subseteq A\) and \(\bar \delta B \subseteq B\). Let \(p(\bar x)\) be a complete \(L^{\bar \delta}\)-type over \(A\) (in \(n\) variables). Let \(q \in S_{L}^{J}(B)\) be some extension of \(p_{J}\). If \[q \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}_{A} B,\] then \(q\) is \(\bar \delta\)-compatible (see Subsection 2.2).
Proof. W.l.o.g., \(A\) and \(B\) are also subrings of \(K\). Let \(\bar a\in K^{n}\) be a realization of \(p\) and \(\bar b\in K^{J}\) be a realization of \(q\). Since \(\mathop{\mathrm{Jet}}(\bar a)\) and \(\bar b\) have the same \(L\)-type \(q\) over \(A\), there exists \(\phi\) an \(L\)-automorphism of \(\mathbb{K}\) fixing \(A\) point-wise and mapping \(\bar b\) to \(\mathop{\mathrm{Jet}}(\bar a)\).
Define \(\bar\varepsilon\mathrel{\vcenter{:}}= \phi^{-1} \circ \bar \delta\circ \phi\). Let
\(\bar\delta_{0}\) be the restriction of \(\bar\delta\) to \(A\),
\(\bar\delta_1\) be the restriction of \(\bar\delta\) to \(B\),
\(\bar\delta_{2}\) be the restriction of \(\bar\varepsilon\) to \(A[\bar b]\).
Notice that \(\bar\varepsilon\) is a tuple of derivations on \(K\) which extends \(\bar \delta_{0}\) and such that \(\bar\varepsilon\) commutes if \(\bar \delta\) commutes. Thus, by the Amalgamation Lemma, there exists a \(k\)-tuple \(\bar\lambda\) of derivations on \(K\) which extends \(\bar\delta_0\), \(\bar\delta_{1}\), \(\bar\delta_{2}\) and such that \(\bar\lambda\) commutes if \(\bar\delta\) commutes. Thus, \(\langle \mathbb{K}, \bar\lambda \rangle \models T^{\bar \delta,?}\) (\(\bar\lambda\) is not generic, in general). Let \(\bar b_0 \in K^{n}\), interpreted in \(\langle \mathbb{K}, \bar\lambda \rangle\), we have that \(\mathop{\mathrm{Jet}}( \bar b_0) = \bar b\), and therefore \(\mathop{\mathrm{Jet}}( \bar b_0)\) satisfies \(q\). Thus, the partial \(L^{\bar \delta}(B)\)-type \[u(\bar x) \mathrel{\vcenter{:}}= T^{\bar \delta,?}_g\cup \mathop{\mathrm{Diag}}_{L^{\bar \delta}}(B) \cup q(\mathop{\mathrm{Jet}}(\bar x))\] is consistent. Since \(\langle \mathbb{K}, \bar \delta \rangle\) is existentially closed and \(\lvert B\rvert^{+}\)-saturated, there exists \(\bar c\in K^{n}\) satisfying \(u\). ◻
It is worth noting that, in view of Lemma 20, the theory \(T^{\bar\delta}\) with respect to \((T, \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}})\) fits into the framework of derivation-like expansions introduced in [@SanchM:24].
Indeed, Lemma 20 yields the amalgamation property required in part (1) of the definition of derivation-like expansions. Moreover, since \(\mathbb{K}\) is algebraically bounded, algebraic closure defines an independence relation, and, in characteristic \(0\), derivations extend uniquely to algebraic extensions. This immediately implies part (2) of the definition.
Furthermore, when \(T = \mathrm{Th}(\mathbb{K})\) is simple, forking independence implies algebraic independence, and hence \(T^{\bar\delta}\) is derivation-like also with respect to \((T, \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}})\).
As a consequence, several of our results can alternatively be obtained from the general theory developed in [@SanchM:24], and our Theorems 21, 22, 28, 30 (as well as Corollary 2) can be recovered from their corresponding general results in that setting.
Nevertheless, we have chosen to include self-contained proofs of these results. This is partly for the convenience of the reader, and partly to maintain a uniform approach and notation throughout the paper, avoiding the need to repeatedly translate between our setting and the abstract framework of [@SanchM:24].
In this section we examine the relations between algebraic closure, independence relations, and (geometric) elimination of imaginaries. The results in this section will be used in [sec:simple] [sec:ddim].
We fix \(\langle \mathfrak C; \bar \delta \rangle\) monster model of \(T^{\bar \delta,?}_g\).
Let \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) be some ternary relation on subsets of \(\mathfrak C\). We define the following ternary relation on subsets of \(\mathfrak C\). \[A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\bar \delta}}\,\,\,\,}}_{C} B \iff \mathop{\mathrm{Jet}}(A) \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{\mathop{\mathrm{Jet}}(C)} {\mathop{\mathrm{Jet}}(B)}.\] Notice that we are not assuming that \(T\) eliminates imaginaries, or that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) extends to \(\mathfrak C^{eq}\)).
Theorem 22. Assume that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is a real-strict independence relation on \(\mathfrak C\) (see 5). Then, \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\bar \delta}}\,\,\,\,}}\) is a real-strict independence relation on \(\langle \mathfrak C; \bar \delta \rangle\).
If moreover \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is real-canonical, then \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\bar \delta}}\,\,\,\,}}\) is real-canonical.
Proof. In [@Adler], a list of axioms for independence relations is presented. The author subsequently demonstrates that the Extension Axiom is equivalent to the conjunction of the Existence Axiom and the Symmetry Axiom. Notably, among the axioms for independence (\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\bar \delta}}\,\,\,\,}}\)) outlined in [@Adler], only the Existence Axiom requires non-trivial verification. Thus, let \(\bar a, \bar b, \bar c\) be small tuples in \(\mathfrak C\). Since \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is an independence relation, there exists a small tuple \(\bar d\subset \mathfrak C\) s.t. \(\bar d\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{\mathop{\mathrm{Jet}}(\bar c)} \mathop{\mathrm{Jet}}( \bar b)\). Let \(p \mathrel{\vcenter{:}}= \mathop{\mathrm{tp}}^{L^{\bar \delta}}(\mathop{\mathrm{Jet}}(\bar a)/ \mathop{\mathrm{Jet}}(\bar c))\) and \(q \mathrel{\vcenter{:}}= \mathop{\mathrm{tp}}^{L}(\bar d/ \mathop{\mathrm{Jet}}(\bar c) \mathop{\mathrm{Jet}}(\bar b))\). By assumption, \(q\) is a complete \(L\)-type extending \(p\), and \[q \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{\mathop{\mathrm{Jet}}(\bar c)} \mathop{\mathrm{Jet}}(\bar b).\] Since \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is real-strict, 12 implies that \[q \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}_{\mathop{\mathrm{Jet}}(\bar c)} \mathop{\mathrm{Jet}}(\bar b)\] and therefore, by the Extension Theorem, there exists \(\bar a' \subset \mathfrak C\) s.t. \(\mathop{\mathrm{Jet}}(\bar a')\) realizes \(q\): hence, \[\bar a' \equiv^{L^{\bar \delta}}_{\bar c} \bar a\ \wedge\ \bar a' \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\bar \delta}}\,\,\,\,}}_{\bar c} \bar b,\] proving that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\bar \delta}}\,\,\,\,}}\) satisfies Existence.
The fact that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\bar \delta}}\,\,\,\,}}\) is real-strict is clear.
Finally, assume that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is real-canonical. Assume that \[\bar c\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\bar \delta}}\,\,\,\,}}_{E\bar b} \bar a\ \wedge\ \bar c\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\bar \delta}}\,\,\,\,}}_{E\bar a} \bar b.\] We have to prove that \[\bar c\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\bar \delta}}\,\,\,\,}}_{\mathop{\mathrm{\mathop{\mathrm{acl}}^{\bar \delta}}}(E \bar a) \cap \mathop{\mathrm{\mathop{\mathrm{acl}}^{\bar \delta}}}(E \bar b)} \bar a\bar b.\] W.l.o.g., we may assume that \(E = \mathop{\mathrm{\mathop{\mathrm{acl}}^{\bar \delta}}}E\). Thus, we have \[\mathop{\mathrm{Jet}}(\bar c) \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{E \mathop{\mathrm{Jet}}(\bar b)} \mathop{\mathrm{Jet}}(\bar a) \ \wedge\ \mathop{\mathrm{Jet}}(\bar c) \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{E\mathop{\mathrm{Jet}}(\bar a)} \mathop{\mathrm{Jet}}(\bar b).\] Since \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is real-canonical, we have \[\mathop{\mathrm{Jet}}(\bar c) \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{\mathop{\mathrm{acl}}(E \mathop{\mathrm{Jet}}(\bar b)) \cap \mathop{\mathrm{acl}}(E \mathop{\mathrm{Jet}}(\bar b))} \mathop{\mathrm{Jet}}(\bar b).\] \(\mathop{\mathrm{acl}}(E \mathop{\mathrm{Jet}}(\bar b)) = \mathop{\mathrm{\mathop{\mathrm{acl}}^{\bar \delta}}}(E \bar b)\), and similarly for \(\bar a\), we are done. ◻
In particular, we can choose \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}:= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}\), and obtain that the induced relation \(\makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}\) is a real-strict independence relation on \(\langle \mathfrak C; \bar \delta \rangle\).
Corollary 2 (cf.[@PillayPR:25], [@SanchM:24]). 1) The algebraic closure on \(\langle \mathfrak C; \bar \delta \rangle\) is given by \[\mathop{\mathrm{acl}}(\mathop{\mathrm{Jet}}(A)),\] for every \(A \subseteq \mathfrak C\).3
2) If \(\langle \mathfrak C; \bar \delta \rangle\) has GEI, then it is rosy, and \(\makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}\) coincides with -forking \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}\) (for \(\langle \mathfrak C; \bar \delta \rangle\)).
Proof. Let \(\mathop{\mathrm{\mathop{\mathrm{acl}}^{\bar \delta}}}\) be the algebraic closure according to \(T^{\bar \delta,?}_g\).
1) If \(a \in \mathop{\mathrm{\mathop{\mathrm{acl}}^{\bar \delta}}}B\), then, since \(\makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}\) is an independence relation, \[a \makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}_{B} a,\] and in particular \[a \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}_{\mathop{\mathrm{Jet}}(B)} a\] that is \(a \in \mathop{\mathrm{acl}}(\mathop{\mathrm{Jet}}(B))\).
Conversely, it is clear that \(\mathop{\mathrm{acl}}(\mathop{\mathrm{Jet}}(B))\) must be contained in \(\mathop{\mathrm{\mathop{\mathrm{acl}}^{\bar \delta}}}(B)\).
2) Since \(\makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}\) is a real-strict independence relation on \(\langle \mathfrak C, \bar\delta \rangle\), we have that the latter is real-rosy; if moreover \(\langle \mathfrak C, \bar\delta \rangle\) has GEI, it is rosy (see [@Adler] for definitions and proofs: in particular, [@Adler]). ◻
Proposition 23. Let \(K\) be any field of characteristic \(0\). Define \(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C} B\) if, for every \(X \subseteq A\) which is algebraically independent over \(C\) (in the field-theoretic sense), \(X\) remains algebraically independent over \(BC\). Then, \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is real-canonical.
Proof. Let \(\tilde{K}\) be the field-theoretic algebraic closure of \(K\). Given \(C \subseteq K\), let \(\tilde{C}\) be the field-theoretic algebraic closure of \(C\) inside \(\tilde{K}\), and \(\mathop{\mathrm{acl}}(C)\) be the field-theoretic algebraic closure of \(C\) inside \(K\).
Let \(D \mathrel{\vcenter{:}}= C_{1} \cap C_{2}\).
Claim 1. \[\tilde{C_{1}} \cap \tilde{C_{2}} = \widetilde{\mathop{\mathrm{acl}}(C_{1}) \cap \mathop{\mathrm{acl}}(C_{2})}.\]
First, w.l.o.g. we may assume that \(C_{i} = \mathop{\mathrm{acl}}(C_{i})\), for \(i = 1, 2\). It is clear that \[\tilde{D} \subseteq \tilde{C_{1}} \cap \tilde{C_{2}};\] we have to prove the opposite inclusion. Let \(C\) be either \(C_{1}\) or \(C_{2}\). Let \(a \in \tilde{C}_{1} \cap \tilde{C}_{2}\) and \(a_{1}, \dotsc, a_{\ell}\) be the conjugates of \(a\) over \(K\). Then, \(a_{j} \in \tilde{C}\), \(j= 1, \dotsc, \ell\). Let \(\sigma_{t}(x_{1}, \dotsc, x_{\ell})\) be one of the symmetric polynomials: we have that \(\sigma_{t}(a_{1}, \dotsc, a_{\ell}) \in \tilde{C} \cap K = C\). Thus, if \(p(x) \in K[x]\) is the minimal polynomial of \(a\) over \(K\), we have \(p \in C[x]\), and therefore \(p \in D[x]\).
Thus, \(a \in \tilde{D}\), proving the claim.
Let \(\tilde{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}}\) be the analogue of \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) inside \(\tilde{K}\).
Claim 2. \(\tilde{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}}\) is real-canonical.
In fact, \(\tilde{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}}\) coincides with \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) on \(\tilde{K}\) seen as a pure field, and, since \(\tilde{K}\) (using the fact that it is an algebraically closed field) is stable with EI, \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) is real-canonical: see [@Adler].
Assume now that \(C_{i} \subseteq B\) for \(i = 1, 2\), and \(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{C_{i}} B\) and \(C_{i} \subseteq B\), for \(i = 1, 2\).
We have \(A \tilde{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}}_{C_{i}} B\) for \(i = 1, 2\). Since \(\tilde{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}}\) is real-canonical, we have that \[A \tilde{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}}_{\tilde{C}_{1} \cap \tilde{C}_{2}} B,\] and therefore, by 1, \[A \tilde{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}}_{\tilde{D}} B,\] which is equivalent to \(A \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{D} B\). ◻
Corollary 3. \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}\) is real-strict and real-canonical.
\(\makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}\) is also real-strict and real-canonical.
Thus, by 1, we have:
Theorem 24. T.f.a.e.:
\(\mathfrak C\) has GEI;
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}\) is strict;
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}\);
\(\mathfrak C\) is super-rosy of -rank 1.
Theorem 25. T.f.a.e.:
\(\langle \mathfrak C; \bar \delta \rangle\) has GEI;
\(\makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}\) is strict;
\(\makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}\) is equal to -forking on \(\langle \mathfrak C; \bar \delta \rangle\).
Conjecture 26. Assume that \(\mathfrak C\) is has GEI. Then, \(\langle \mathfrak C; \bar \delta \rangle\) also has GEI.
Proposition 27. Let \(\mathfrak C\) be a monster model of some theory. Assume that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is an independence relation on \(\mathfrak C\) satisfying the following conditions:
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is real-canonical;
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is “controlled by a rank”: there exist a linearly ordered set \(\langle I ; \leq \rangle\) and a function \(\rho\) associating an element of \(I\) to each type (over some small subset of the monster model \(\mathfrak C\) in finitely many real variables), such that:
\(\rho\) is invariant under automorphisms,
\(\rho\) for every \(\bar a\), \(B\), \(C\), \(\rho(\bar a/B) \leq \rho(a/BC)\), with equality iff \(\bar a\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}_{B} C\),
for every non-empty set \(X\) which is type-definable over \(A\), \(\rho(\bar x/A)\) attains a maximum on \(X\) (e.g., \(\rho\) is upper semi-continuous).
Let \(e \in \mathfrak C^{eq}\), where \(e = f(\bar a)\) for some (real finite tuple) \(\bar a\in \mathfrak C^{\ell}\) and some function \(f\) which is definable without parameters. Let \(E \coloneq \mathop{\mathrm{acl^{eq}}}(e) \cap \mathfrak C\) and \(P\) be the set of realization of \(\mathop{\mathrm{tp}}(a/E)\) (inside \(\mathfrak C\)). Then there exists \(\bar c\in P\) such that \(f (\bar c) = e\) and \(\bar a\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{E} \bar c\).
Proof. 27 follows the proof in [@HC:99] and [@MS]. As noticed in [@HC:99], Neumann’s Lemma implies that there exists \(\bar b\) conjugate of \(\bar a\) over \(E \cup \{e\}\) such that \[\label{eq:1} \mathop{\mathrm{acl}}(E\bar a) \cap \mathop{\mathrm{acl}}(E\bar b) = \mathop{\mathrm{acl^{eq}}}(E e) \cap \mathfrak C= E\tag{1}\] (see [@EH:93]). Let \(Q\) be the set of \(\bar b\in P\) satisfying 1 .
Claim 3. \(Q\) is type-definable over \(E \bar a\). 4
Since \(P\) is type-definable, it suffices to show that the set \[X \mathrel{\vcenter{:}}= \{\bar x\in \mathfrak C^{\ell}: \mathop{\mathrm{acl}}(E\bar a) \cap \mathop{\mathrm{acl}}(E\bar x) \supsetneq E\}\] is \(\mathord{\vee}\)-definable. For every \(d \in \mathop{\mathrm{acl}}(E\bar a) \setminus E\), let \(X_{d} \mathrel{\vcenter{:}}= \{\bar x\in \mathfrak C^{\ell}: d \in \mathop{\mathrm{acl}}(E \bar x)\}\). \(X_{d}\) is \(\mathord{\vee}\)-definable(over \(E d\)) because it is a union of sets defined by formulae of the form \(\alpha(\bar c,d, \bar x) \ \wedge\ \lvert\alpha(\bar c, \mathfrak C, x)\rvert \leq n\), as \(\alpha\) varies among all \(L\)-formulae and \(n\) varies in \(\mathbb{N}\). Therefore, \(X = \bigcup_{d \in \mathop{\mathrm{acl}}(E\bar a) \setminus E} X_d\) is also \(\mathord{\vee}\)-definable(over \(\mathop{\mathrm{acl}}(E \bar a)\)).
By assumption, there exists \(\bar b\in Q\) such that \(\rho(\bar b/E \bar a)\) is maximum (inside \(Q\)).
Let \(\bar c\models \mathop{\mathrm{tp}}(\bar b/E\bar a)\) such that \(\bar a\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{E \bar a} \bar b\). As in [@HC:99] (cf.[@MS]), we see that \[\bar c\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{E \bar b} \bar a\ \wedge\ \bar c\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{E \bar a} \bar b.\] Since \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) is real-canonical, and \(\mathop{\mathrm{acl}}(E \bar a) \cap \mathop{\mathrm{acl}}(E \bar b) = E\), we have that \[\bar c\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{E} \bar a\bar b\] and therefore \(\bar c\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{E} \bar a\), proving 27. ◻
We will now prove that if T is simple, then \(T^{\bar \delta,?}_g\) is also simple. Our proof relies on the crucial notion of “Independence over Algebraically Closed Substructures” (IACS), which we define and explore in this section.
Theorem 28 (Simplicity of \(T^{\bar \delta,?}_g\)). If \(T\) is simple, then \(T^{\bar \delta,?}_g\) is also simple.
The above is non-trivial, but thanks to the preliminary work we can give an easy proof. We fix \(\langle \mathfrak C; \bar \delta \rangle\) monster model of \(T^{\bar \delta,?}_g\). Assume that \(\mathfrak C\) is simple. Let \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) be Shelah-forking relation on \(\mathfrak C\), and \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f,\delta}}\,\,\,\,}}\) be the induced independence relation on \(\langle \mathfrak C; \bar \delta \rangle\) as in 6.
Since we want to prove a stronger version of 28, we need the following definition.
Definition 8. An independence relation \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}\) satisfies “Independence over Algebraically Closed Substructures” (IACS) if:
For every \(\bar a, \bar b\in \mathfrak C^{n}\), \(\bar a', \bar b' \in \mathfrak C^{m}\), for every \(M \subset \mathfrak C\) with \(M\) small and algebraically closed, if \[\bar a\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{M} \bar b, \qquad \bar a' \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{M} \bar a, \qquad \bar b' \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{M} \bar b, \qquad \bar a' \equiv^{L}_{M} \bar b'\] then there exists \(\bar c\in \mathfrak C^{m}\) such that \[\bar c\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{M} \bar a\bar b, \qquad \bar c\equiv^{L}_{M \bar a} \bar a', \qquad \bar c\equiv^{L}_{M \bar b} \bar b'.\]
We say that \(T\) satisfies IACS if \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) satisfies it.
As we will see later, IACS is closely related to (weak) elimination of imaginaries cf.[@HC:99].
Remark 29. Obviously, IACS implies Independence over Models; the converse is not true: for instance, let \(L = \{E\}\), where \(E\) is a binary relation, and \(T\) be the theory saying that \(E\) has two equivalence classes, both infinite; then, \(T\) is \(\omega\)-stable and hence simple, therefore \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) satisfies Independence over Models, but it is easy to see that it does not satisfy IACS (take \(M = \emptyset\)). Indeed it is enough to consider four distinct elements \(a, a', b, b'\) with \(a', b'\) in two different equivalence classes.
Theorem 30. Assume that \(T\) is simple. Then, \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f,\delta}}\,\,\,\,}}\) is a real-strict independence relation on \(\langle \mathfrak C, \bar\delta \rangle\) and it satisfies Independence over Models.
Thus, \(T^{\bar \delta,?}_g\) is simple, and \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f,\delta}}\,\,\,\,}}\) is Shelah-forking on \(\langle \mathfrak C; \bar \delta \rangle\).
If moreover \(T\) satisfies IACS, then also \(T^{\bar \delta,?}_g\) satisfies IACS.
Proof. From 22 we have that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f,\delta}}\,\,\,\,}}\) is a strict independence relation.
We have to show that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f,\delta}}\,\,\,\,}}\) satisfies Independence over Models (resp., over Algebraically Closed Substructures) if \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) satisfies it. Let \(\bar a\), \(\bar b\), \(\bar a'\), \(\bar b'\) be as above and \(M\) be a small elementary \(L^{\bar \delta}\)-substructure (resp., small \(L^{\bar \delta}\)-algebraically closed substructure) of \(\langle \mathfrak C, \bar\delta \rangle\). The assumptions become: \[\begin{align} \mathop{\mathrm{Jet}}(\bar a) &\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}_{M} \mathop{\mathrm{Jet}}(\bar b), & \mathop{\mathrm{Jet}}(\bar a') &\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{}}\,\,\,\,}}_{M} \mathop{\mathrm{Jet}}(\bar a), \\ \mathop{\mathrm{Jet}}(\bar b') &\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}_{M} \mathop{\mathrm{Jet}}(\bar b), & \mathop{\mathrm{Jet}}(\bar a') &\underset{M}{\equiv^{L^{\bar \delta}}} \mathop{\mathrm{Jet}}(\bar b') \end{align}\]
By 15 (applied to \(\mathfrak C\)), \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) satisfies Independence over Models (resp., by assumption, \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) satisfies IACS), and therefore there exists \(\bar d\in M^{\Gamma_{n}}\) such that \[\bar d\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}_{\mathfrak C} \mathop{\mathrm{Jet}}{(\bar a)} \mathop{\mathrm{Jet}}{(\bar b)}, \qquad \bar d\underset{\mathfrak C\mathop{\mathrm{Jet}}(\bar a)}{\equiv^{L}} \mathop{\mathrm{Jet}}{(\bar a')}, \qquad \bar d\underset{\mathfrak C\mathop{\mathrm{Jet}}(\bar b)}{\equiv^{L}} \mathop{\mathrm{Jet}}({\bar b'})\] Let \(q \mathrel{\vcenter{:}}= \mathop{\mathrm{tp}}^{L}(\bar d/ M \mathop{\mathrm{Jet}}(\bar a) \mathop{\mathrm{Jet}}(\bar b))\), and \(p\) be restriction of \(q\) to \(\mathfrak C\). Notice that \(p\) is realized by \(\delta \bar a\), and it is therefore \(\bar\delta\)-compatible. Moreover, \(\bar d\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}_{\mathfrak C} \mathop{\mathrm{Jet}}({\bar a}) \mathop{\mathrm{Jet}}({\bar b})\) and therefore \(q \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}_{\mathfrak C} \mathop{\mathrm{Jet}}(\bar a) \mathop{\mathrm{Jet}}(\bar b)\), and therefore \[q \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}_{\mathfrak C} \mathop{\mathrm{Jet}}(\bar a) \mathop{\mathrm{Jet}}(\bar b).\] Thus, by the Extension Theorem, \(q\) is \(\bar\delta\)-compatible, and it is therefore realized by \(\mathop{\mathrm{Jet}}(\bar c)\) for some \(\bar c\in \mathfrak C^{m}\). Thus, we have \[\mathop{\mathrm{Jet}}(\bar c) \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}_{\mathfrak C} \mathop{\mathrm{Jet}}({\bar a}) \mathop{\mathrm{Jet}}({\bar b}), \quad \mathop{\mathrm{Jet}}(\bar c) \underset{\mathfrak C\mathop{\mathrm{Jet}}(\bar a)}{\equiv^{L}} \mathop{\mathrm{Jet}}{(\bar a')}, \quad \mathop{\mathrm{Jet}}(\bar c) \underset{\mathfrak C\mathop{\mathrm{Jet}}(\bar b)}{\equiv^{L}} \mathop{\mathrm{Jet}}{(\bar b')}\] which implies the conclusion. ◻
Theorem 31. Assume that \(T\) is algebraically bounded, simple, with GEI and \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) is real-canonical (e.g.\(T\) is either stable or supersimple). Then:
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\);
\(T\) is super-simple of \(SU\)-rank 1;
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}\) is strict and real-canonical;
Shelah forking on \(T^{\bar \delta,?}_g\) is equal to \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f,\delta}}\,\,\,\,}}\) and is also strict and real-canonical;
\(T^{\bar \delta,?}_g\) has GEI.
Proof. \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}\) by 18 and it is therefore strict. \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) by 19 \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}\) is real-canonical by 3.
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f,\delta}}\,\,\,\,}}\) is Shelah’s forking by 30; it is therefore strict; it is real-canonical by 22.
\(T^{\bar \delta,?}_g\) has GEI by 3) and 11. ◻
By Theorem A, we can deduce that the assumptions of 31 can be weakened to “T is algebraically bounded and simple”.
In this Section we give some sufficient conditions for \(T^{\bar \delta,?}_g\) to have Elimination of Imaginaries (EI) when \(T\) is simple (see 34 4); for previous results see [@MS; @Mohamed].
Given a monster model \(\mathfrak C\), we use the following convention:
\(\bar a, \bar b, \bar c, \bar d\) are finite tuples of real elements, \(A, B,C, D, E\) are small sets of real elements, while \(e\) will be an imaginary
element. \(\mathop{\mathrm{acl}}\) will always denote the algebraic closure inside \(\mathfrak C\) (and not \(\mathfrak C^{eq}\)); when we consider two
theories, \(T\) and \(T^{\bar \delta,?}_g\), we will use \(\mathop{\mathrm{acl}}\) for the algebraic closure according to \(T\), and \(\mathop{\mathrm{\mathop{\mathrm{acl}}^{\bar \delta}}}\) for the algebraic closure according to \(T^{\bar \delta,?}_g\).
Definition 9. We say that \(T\) is “good” if:
\(T\) is algebraically bounded,
\(T\) has EI,
\(T\) is supersimple.
Remark 32. If \(T\) is good then \(T\) has EHI and \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}\) (13 24).
Moreover conditions 1) and 2) imply that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}\) (24).
Conditions 2) and 3) imply that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) satisfies IACS(16).
Finally, if \(T\) is good, then \(T\) is supersimple of SU rank 1.
Remark 33. If we consider the conditions 1) and 2) of Definition 9 together with the assumption that the theory \(T\) is stable, then it is easy to see that \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{\th}}\,\,\,\,}}= \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}\) which implies that \(T\) is superstable of SU rank 1 (a particular case of supersimplicity), and therefore it is good. By Theorem A, we can further weaken the assumption to conditions 1) and 2) plus \(T\) is simple.
Theorem 34. If \(T\) is good, then \(T^{\bar \delta,?}_g\) is simple, \(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f,\delta}}\,\,\,\,}}\) is real-canonical and satisfies IACS, and \(T^{\bar \delta,?}_g\) has EI.
We will prove the above theorem at the end of this subsection, after some consequences and preliminary results.
Corollary 4. If \(T\) is algebraically bounded and stable with EI, then \(T^{\bar \delta,?}_g\) has EI.
Proof. If \(T\) is stable with EI, then it is superstable (of U-rank 1). (Theorem A gives that \(T = \mathop{\mathrm{ACF_0}}\)). ◻
We prove a more general result that we will use for the proof of Theorem 34.
Proposition 35. Let \(\mathfrak C\) be a monster model satisfying the following conditions:
\(\mathfrak C\) is simple;
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) satisfies IACS;
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) is real-canonical;
\(\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}\) is controlled by a rank \(\rho\), as defined in [en:controlled-rank] of 27.
Then, \(\mathfrak C\) has Weak Elimination of Imaginaries (WEI). If moreover \(\mathfrak C\) expands a field, then \(\mathfrak C\) has EI.
Proof. The proof keeps following the one in [@HC:99] and [@MS],5. Here are some more details.
Let \(e\) be an imaginary element. Write \(e = f(\bar a)\) for some (real finite tuple) \(\bar a\in \mathfrak C^{\ell}\) and some function \(f\) which is definable without parameters. Let \(E := \mathop{\mathrm{acl^{eq}}}(e) \cap \mathfrak C\). Our thesis that \(U\) has WEI is equivalent to \(e \in \mathop{\mathrm{dcl^{eq}}}(E)\) (when \(\mathfrak C\) expands a field, WEI implies EI see Remark 7).
Let \(P\) be the set of realization of \(\mathop{\mathrm{tp}}(\bar a/E)\) (inside \(\mathfrak C\)). By 27, there exists \(\bar c\in P\) such that \(f (\bar c) = e\) and \(\bar a\) and \(\bar c\) are (forking)-independent over \(E\).
Claim 4. \(f\) is constant on \(P\).
Otherwise, there exists \(\bar d' \in P\) such that \(f(\bar a) \neq f(\bar d')\). Let \(\bar d'', \bar a''\) be such that \(\bar d'' \bar a'' \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}_{E} \bar d' \bar a\) and \(\bar d'' \bar a'' \equiv_{E} \bar d' \bar a\). Thus, \(f(\bar d'') \neq f(\bar a'')\), \(\bar d'' \equiv_{E} \bar a\), and \(\bar a'' \equiv_{E} \bar a\). Thus, choosing either \(\bar d\mathrel{\vcenter{:}}= \bar d''\) or \(\bar d\mathrel{\vcenter{:}}= \bar a''\) we get \(f(\bar d) \neq f(\bar a)\) and and \(\bar d\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{f}}\,\,\,\,}}_{E} \bar a\).
By IACS(since \(E\) is algebraically closed), (by taking \(\bar a\mathrel{\vcenter{:}}= \bar a\), \(\bar b\mathrel{\vcenter{:}}= \bar d\), \(\bar a' \mathrel{\vcenter{:}}= \bar c\), and \(\bar b' \mathrel{\vcenter{:}}= \bar a\) in 8) there exists \(\bar c' \in \mathfrak C^{\ell}\) such that \(\bar c' \equiv_{E \bar d} \bar a\) and \(\bar c' \equiv_{E \bar a} \bar c\). Since \(e = f(\bar a) \neq f(\bar d)\), we get \(f(\bar c') \neq e\); but since \(e = f(\bar a) = f(\bar c)\), we get \(f(\bar c') = e\), absurd.
Claim 5. \(e \in \mathop{\mathrm{dcl^{eq}}}(E)\).
By 4 and compactness, there exists an \(L(E)\)-definable set \(X\) such that \(P \subseteq X\) and \(f\) is constant on \(X\). Thus, \(e\) is defined by the \(L(E)\)-formula \(\exists \bar x\in X\, f(\bar x) = y\). ◻
Proof of 34. We have to show that \(T\) and \(T^{\bar \delta,?}_g\) satisfy the assumptions of 35.
For \(T\), we define \(\rho(\bar a/B)\) to be the rank given by the algebraic closure (notice that under our assumptions \(T\) is super-simple of rank \(1\), and \(\rho\) is equal to the \(SU\)-rank). Since \(\rho(a_{1}, \dotsc, a_{m}) / B \leq m\), condition [en:USC] of Proposition 27 is automatically true, and therefore \(T\) satisfies the assumptions of 35.
We consider now \(T^{\bar \delta,?}_g\): we have to show that the various properties of \(T\) that we need are inherited by \(T^{\bar \delta,?}_g\).
We have seen before that \(T^{\bar \delta,?}_g\) is simple with IACS and real-canonical.
We need to introduce a suitable function \(\rho\) and show that it satisfies the conditions in 35. We can use the analogue of the dimension function \(\dim_{\mathcal{D}}\) in [@MS]. For every \(r \in \mathbb{N}\), define \(\rho(\bar a/B)(n)\) as the transcendence degree of \(\mathop{\mathrm{Jet}}_n(\bar a)\) over \(\mathop{\mathrm{acl}}(\mathop{\mathrm{Jet}}(B))\), where \[\mathop{\mathrm{Jet}}_n(\bar a) = \{\mu \bar a: \mu \in \Gammaand\lvert\mu \rvert\leq n\}\] Thus, \(\rho(\bar a/B) \in \omega^{\omega}\). If we endow \(\omega^{\omega}\) with the lexicographic ordering, then \(\rho\) is upper semi-continuous, and satisfies the conditions in 35. ◻
This section is based on a suggestion by Elliot Kaplan.
Distality is a stronger condition than NIP, it characterizes NIP theories that are as far from stability as possible. For a formal definition and key properties of distal theories, see [@Simon;
@AscheCGZ-22]. In [@FT:24] we showed that if \(T\) is NIP, then \(T^{\bar \delta,?}_g\) is NIP. A similar results hold when \(T\) is distal.
Theorem 36. If \(T\) is distal, then \(T^{\bar \delta,?}_g\) is distal.
Proof. Immediate from 4 and [@AscheCGZ-22]. ◻
In [@FT:24], we showed that if the theory \(T\) is stable, then \(T^{\bar \delta,?}_g\) retains stability. The analogous result for \(\omega\)-stability is false: here we give necessary and sufficient conditions for \(T^{\bar \delta,?}_g\) to be \(\omega\)-stable. Recall that countable theory is totally transcendental iff the theory is \(\omega\)-stable (see [@Sacks]).
Theorem 37. \(T^{\bar \delta,?}_g\) is totally transcendental iff \(T = \mathop{\mathrm{ACF_0}}\) (i.e., \(T\) is the theory of pure algebraically closed fields, possibly with some constants) and \(\bar\delta\) commute.
We first need the following lemma (cf. Theorem A):
Lemma 38. \(T\) is algebraically bounded and strongly minimal iff \(T = \mathop{\mathrm{ACF_0}}\).
Proof. It is clear that, if \(T = \mathop{\mathrm{ACF_0}}\), then \(T\) is algebraically bounded and strongly minimal.
Conversely, assume that \(T\) is algebraically bounded and strongly minimal. Let \(\mathbb{K}\models T\). Since \(\mathbb{K}\) is strongly minimal, as a field \(K\) is algebraically closed (see Macintyre’s Theorem [@Poizat:groups]). By [@Hrushovski:92], every \(\mathbb{K}\)-definable set is definable in the field language. ◻
Proof of 37. If \(T = \mathop{\mathrm{ACF_0}}\), then \(T^{\bar \delta}_g\) is \(\omega\)-stable [@mcgrail].
Let us now consider the converse. Let \(\langle \mathbb{K}, \bar \delta \rangle\models T^{\bar \delta,?}_g\).
Assume that \(\bar\delta\) do not commute. Let \(C_{1}\) be the fixed field of \(\delta_{1}\). Notice that \(\delta_{2}(C_{1}) = K\), since \(\mathbb{K}\) is existentially closed. Thus, \(\mathop{\mathrm{MR}}(C_{1}) = \mathop{\mathrm{MR}}(K)\). However, \(C_{1}\) is a subgroup of \(K\) of infinite index, and therefore \(\mathop{\mathrm{MR}}(K) = \infty\), proving that \(\langle \mathbb{K}, \bar \delta \rangle\) is not \(\omega\)-stable.
We have to prove that \(T = \mathop{\mathrm{ACF_0}},\) so by Lemma 37 we assume that \(K\) is not strongly minimal. We want to show that \(\langle \mathbb{K}, \bar \delta \rangle\) is not \(\omega\)-stable. It suffices to show that, assuming that \(L(K)\) is countable, there exist \(2^{\omega}\) \(L^{\bar \delta}\)-1-types over \(K\). Then, let \(X \subset K\) which is \(\mathbb{K}\)-definable and such that both \(X\) and \(\mathbb{K}\setminus X\) are infinite. Fix \(J \subseteq \mathbb{N}\) and consider the following partial \(L^{\bar \delta}(K)\)-type. \[p_{J}(x) = \bigl( \delta_{1}^{i} x \in X: i \in J \bigr) \cup \bigl( \delta_{1}^{i} x \notin X: i \notin J \bigr).\] Notice \(p_{J}\) is indeed a partial type and, for every \(J \neq J'\), \(p_{J}\) and \(p_{J'}\) are incompatible. Thus, \(\lvert S^{1}_{L^{\bar \delta}}(K)\rvert \geq 2^{\omega}\).
◻
We give some interesting results on the field of constants. Before we introduce some notations and definitions.
Definition 10. The field of constants is the set \[\mathcal{C}_{\bar \delta}\mathrel{\vcenter{:}}= \{a \in K: \delta_{1}(a) = \dots = \delta_{k}(a) = 0\}\]
We consider the reduct \(\langle \mathbb{K},\mathcal{C}_{\bar \delta} \rangle\) (that is, the expansion of \(\mathbb{K}\) with a unary predicate for \(\mathcal{C}_{\bar \delta}\)).
Observe that \(\mathcal{C}_{\bar \delta}\) is dense in \(\mathbb{K}\) w.r.t. the matroid \(\mathop{\mathrm{acl}}\): that is, for every \(Z \subseteq K\) which is \(L\)-definable (remember that \(L\) is the signature of \(\mathbb{K}\)) with parameters in \(\mathbb{K}\) and large, \(Z\) intersects \(\mathcal{C}_{\bar \delta}\); moreover, \(\mathcal{C}_{\bar \delta}\) is also L-algebraically closed in \(\mathbb{K}\). Thus, \(\langle \mathbb{K},\mathcal{C}_{\bar \delta} \rangle\) is a lovely pair of geometric structures (in the sense of [@BerensteinV:10; @Boxall]: see also [@Fornasiero:matroid]). Thus, we can apply the known results (see [@BerensteinV:10; @Boxall; @Fornasiero:matroid]).
Lemma 39. \(\bar b\in K^{\ell}\) and \(B := \mathop{\mathrm{dcl}}_L(\mathop{\mathrm{Jet}}(\bar b)= \mathop{\mathrm{dcl}}_{L^{\bar \delta}}(\mathop{\mathrm{Jet}}(\bar b))\). Then, \(B\) is P-independent: that is, \(B \mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M}}\,\,\,\,}}_A \mathcal{C}_{\bar \delta}\), where \(A := B \cap P\).
Proof. Assume not: then, there exists \(\bar d \in \mathop{\mathrm{Jet}}(\bar b)^n\) s.t.\(\bar d\) is L-algebraically independent over \(A\) but the following system has a solution \(\bar c\in (K^n)^m\): \[\label{eq:d-system} \left\{\begin{align} \delta \bar x&= 0\\ \bar x&\neq 0\\ \sum_{i,j} x_{ij}d_j^i &= 0. \end{align}\right.\tag{2}\] W.l.o.g., we may assume that \(\langle \mathbb{K}, \bar\delta \rangle\) is a monster model. Let \(\bar b' \equiv_A^{L^{\bar \delta}} \bar b\) s.t. \(\bar b' \makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}_A \bar c\). Let \(\sigma\) be an \(L^{\bar \delta}\)-automorphism of \(\mathbb{K}\) over \(A\) mapping \(\bar b\) to \(\bar b'\) and \(\bar d' := \sigma(\bar d)\). Then, \(\bar d'\) is \(L\)-algebraically independent over \(A\) and therefore 2 (with \(\bar d'\) in place of \(\bar d\)) cannot have a solution in \(K\). But \(\bar b\) and \(\bar b'\) have the same \(L^{\bar \delta}\)-type, absurd. ◻
Theorem 40. \(\mathcal{C}_{\bar \delta}\) is an elementary \(L\)-substructure of \(\mathbb{K}\).
Let \(\bar b\in K^{\ell}\) and \(X \subseteq \mathcal{C}_{\bar \delta}^{n}\) be \(L^{\bar \delta}\)-definable with parameters \(\bar b\). Then, there exists \(Y \subseteq K^{n}\) which is \(L\)-definable (in \(\mathbb{K}\)) with parameters \(\mathop{\mathrm{Jet}}(\bar b)\) such that \(X = Y \cap \mathcal{C}_{\bar \delta}^{n}\). If moreover \(\bar b\in \mathcal{C}_{\bar \delta}\), then \(X\) is \(L\)-definable in \(\mathcal{C}_{\bar \delta}\) with parameters \(\bar b\): equivalently, there exists \(Y \subseteq K^{n}\) which is \(L\)-definable in \(\mathbb{K}\) with parameters \(\bar b\) such that \(X = Y \cap K^{n}\).
Proof. The Theorem is a reformulation of [@Fornasiero:matroid] (see also [@BerensteinV:10]), using 39. ◻
Notice that, by 2, \(\mathcal{C}_{\bar \delta}\) is algebraically closed in \(\mathbb{K}\) w.r.t. the \(L^{\bar \delta}\)-structure.
Definition 11. A basic formula is a formula of the form \[\exists \bar y\, \bigl( \bar y\in K^{\ell} \wedge \psi(\bar x, \bar y) \bigr)\] where \(\psi\) is an \(L\)-formula. A basic set is a set definable by a basic formula (with parameters from \(\mathbb{K}\)).
Theorem 41. Let \(Z \subseteq K^{n}\) be definable in \(\langle \mathbb{K},\mathcal{C}_{\bar \delta} \rangle\) with parameters from \(\mathbb{K}\). Then, \(Z\) is a finite Boolean combination of basic sets, with the same parameters as \(Z\).
The Theorem is a reformulation of [@BerensteinV:10] and [@Fornasiero:matroid].
Remark 42. Let \(\langle A,B \rangle\) be a lovely pair of geometric structures, with \(A \models T\). Then, there exists \(B^{*} \succeq B\) and a derivation \(\delta^*\) on \(B^{*}\) such that \(\langle B^{*}, \delta^{*} \rangle \models T^{\delta}_g\) and \(\langle B^{*}, A^{*} \rangle \succeq \langle B,A \rangle\), where \(A^{*} \mathrel{\vcenter{:}}= \mathfrak C_{\delta^{*}}\)
Proof. The theory \(T^{lovely}\) of lovely pairs of models of \(T\) is complete (see [@BerensteinV:10]). Let \(\langle B^{*}, A^{*} \rangle \succeq \langle B,A \rangle\) be a 0-big (a.k.a. “splendid”: see [@Hodges]). By bigness, there exists a derivation \(\delta^{*}\) on \(B^{*}\) satisfying the conclusion. ◻
[@KP] use a particular case of the above remark to (re-)prove some results about lovely pairs.
For more results (in particular on imaginaries in \(\langle \mathbb{K},\mathcal{C}_{\bar \delta} \rangle\)) see [@BerensteinV:10; @Boxall; @Fornasiero:matroid].
The open core of a topological structure is the structure generated by the sets which are both open and definable. We investigate expansions of algebraically bounded topological structures with generic derivations: more precisely we prove that, under very weak assumptions, \(T\) is the open core of \(T^{\bar \delta,?}_g\) (5).
By a topology \(\tau\) on \(T\) we mean an assignment, to every \(\mathbb{K}\models T\) and every \(n \in \mathbb{N}\), of a topology \(\tau_{n, \mathbb{K}}\) on \(K^{n}\).
Definition 12 ([@Pillay:87]). \(\tau\) is definable if, for every \(n\in \mathbb{N}\), there exists an \(L\)-formula (without parameters) \(\alpha_{n}(\bar x,\bar y)\), where \(\bar x\) is an \(n\)-tuple and \(\bar y\) is a finite tuple, such that, for every \(\mathbb{K}\models T\), we have that \[\{\alpha(\mathbb{K},\bar b): \bar b\in K^{\lvert\bar y\rvert}\}\] is a basis of open sets for \(\tau_{n, \mathbb{K}}\). The family \(\{\alpha(\mathbb{K},\bar b): \bar b\in K^{\lvert\bar y\rvert}\}\) is a definable basis for the topology.
If we give only a topology on \(K\), we assume that the topology on \(K^{n}\) is the product topology.
Definition 13. \(T\) is an open core of \(T^{\bar \delta,?}_g\) if, for every \(\langle \mathbb{K}, \bar \delta \rangle\models T^{\bar \delta,?}_g\) and every \(n \in \mathbb{N}\), every \(\tau\)-open \(L^{\bar \delta}(K)\)-definable subset of \(K^{n}\) is already \(L(K)\)-definable.
We give: sufficient conditions for \(T\) being an open core of \(T^{\bar \delta,?}_g\) (see 45 47) and for \(T^{\bar \delta,?}_g\) to have Elimination of Imaginaries (see 48).
In an article in preparation we will present another approach, with slightly different assumptions and a very different proof.
Let \(\mathbb{K}\models T\) and \(n \in \mathbb{N}\). Let \(X \subseteq K^{n}\) be an \(L(K)\)-definable set.
Definition 14. Given \(\bar a\in K^{n}\), the Local Dimension of \(X\) at \(\bar a\) is \[\mathop{\mathrm{loc-dim}}_{\bar a}(X) \mathrel{\vcenter{:}}= \min\{ \dim(X \cap U): U \text{ open definable set containing \bar a}\}.\] The Local Dimension of \(X\) is \[\mathop{\mathrm{loc-dim}}(X) \mathrel{\vcenter{:}}= \max\{ \mathop{\mathrm{loc-dim}}_{\bar a}(X): \bar a\in X \}.\] We say that \(X\) has constant local dimension \(d\) if, for every \(\bar a\in X\), \(\mathop{\mathrm{loc-dim}}_{\bar a}(X) = d\).
Remark 43. \(\mathop{\mathrm{loc-dim}}(X) \leq \dim(X)\).
Definition 15. \(\dim\) is local (w.r.t. to \(\tau\)) if, for every \(\mathbb{K}\models T\), for every \(n \in \mathbb{N}\), and every \(X \subseteq K^{n}\) \(L(K)\)-definable, \(\dim(X) = \mathop{\mathrm{loc-dim}}(X)\).
[@FH:12] gives sufficient conditions for dim to be local. It gives also an example where it is not local (the Sorgenfrey plane).
Fact 44 ([@FH:12]). Assume that \(\tau\) is definable, the topology on \(K^{n}\) is the product topology (for every \(n \in \mathbb{N}\)), \(\langle K,+ \rangle\) is a Hausdorff topological group (for every \(\mathbb{K}\models T\)), and every open definable set is large, i.e. it has the same dimension of ambient space.
Then, \(\dim\) is local.
Assumptions 45. We assume that, for every \(n,m \in \mathbb{N}\) and \(\mathbb{K}\models T\),
\(\tau\) is definable;
the projection map \(K^{n} \times K^{m} \to K^{n}\) is continuous;
\(\tau_{n, \mathbb{K}}\) is invariant under permutation of coordinates;
\(\dim\) is local.
Examples 46. The following topologies satisfy the assumptions.
\(T =\mathop{\mathrm{RCF}}\) with the usual Euclidean topology.
\(T\) is a theory of Henselian valued fields with the valuation topology.
Theorem 47 (Open Core). \(T\) is the open core of \(T^{\bar \delta,?}_g\).
More precisely: let \(\langle \mathbb{K}, \bar \delta \rangle\models T^{\bar \delta,?}_g\) and \(A \subseteq K\) s.t. \(\bar\delta(A) \subseteq A\). Let \(X \subseteq K^{n}\) be \(L^{\bar \delta}(A)\)-definable and closed. Then, \(X\) is \(L(A)\)-definable.
We give a proof later in Section 11.1. Particular cases of 47 were already known: see [@KP].
Theorem 48. Assume moreover that \(\tau\) satisfies the following condition:
Then, \(T^{\bar \delta,?}_g\) has Elimination of Imaginaries modulo \(T^{eq}\).
Proof. With trivial modifications, the proof suggested by M. Tressl in [@FK] works. ◻
For the following corollary, we spell all conditions explicitly.
Corollary 5. Let \(T\) be algebraically bounded, with a definable topology \(\tau\). Assume that \(\tau\) is a non-trivial and non-discrete ring topology. Then, \(T\) is the open core of \(T^{\bar \delta,?}_g\).
If moreover \(\tau\) and satisfies Assumption [top:boundary] in 48, then \(T^{\bar \delta,?}_g\) has Elimination of Imaginaries modulo \(T^{eq}\).
Proof. Since \(\tau\) is a non-trivial ring topology, it is Hausdorff (see e.g.[@Warner:89]).
We only need to show that \(\dim\) is \(\tau\)-local.
By 44, it suffices to show that every nonempty open subset of \(K^{n}\) is large.
For \(n=1\), since \(\tau\) is Hausdorff and non-discrete, every nonempty open subset of \(K\) is infinite and hence large.
For \(n > 1\), let \(U \subseteq K^{n}\) be open and nonempty. Let \(B_{1}, \dotsc, B_{n} \subseteq K\) be open and nonempty s.t. \(B \mathrel{\vcenter{:}}= B_{1} \times \dots \times B_{n} \subseteq K^{n}\). Since, by the case \(n=1\), each \(B_{i}\) is large, then \(B\) is large. ◻
We present some preliminary results before giving the proof.
Fix \(n \in \mathbb{N}\) and \(\bar x= \langle x_{1}, \dotsc, x_{n} \rangle\). We use the same notation as in 2.
We need to endow \(K^{\Gamma_{n}}\) with a topology. For every \(J\) finite subset of \(\Gamma_{n}\) of cardinality \(m \in \mathbb{N}\), \(\tau_{m, \mathbb{K}}\) is a topology on \(K^{m}\) and hence on \(K^{J}\) (it does not depend on how we enumerate the elements of \(J\), since we assumed that \(\tau_{m, \mathbb{K}}\) is invariant under permutation of coordinates). We have a natural map \(\Pi_{J}: K^{\Gamma_{n}} \to K^{J}\). We define the topology \(\tau_{\mathbb{K},\Gamma_{n}}\) as the coarsest topology making all the maps \(\Pi_{J}\) continuous: equivalently, a basis of \(\tau_{\mathbb{K}, \Gamma_{n}}\) is given by \[\{\Pi_{J}^{-1}(U): U \subseteq K^{J} \text{ open}, J \subset_{fin} \Gamma_{n}\}.\]
Let \(\mathbb{K}\models T\) be a monster model (i.e., \(\lambda\)-saturated and \(\lambda\)-homogeneous for some sufficiently large cardinal \(\lambda > \aleph_{0} + \lvert L\rvert\)) and \(A \subset K\) be a “small” subset (i.e., \(\lvert A\rvert < \lambda\)).
Let \(J \subseteq \Gamma_{n}\) (\(J\) could be finite or infinite); again, we consider \(J\) as a set of indices and \(x_{J}\) as a tuple of variables (see Section 2.2 )
A partial type \(p(x_{J})\) over \(A\) determines a (type-definable) subset \(Z \subseteq K^{J}\). If \(J\) is finite, we can define the dimension of \(Z\) as \[\dim(Z) \mathrel{\vcenter{:}}= \min \{\dim(\alpha(\mathbb{K}): \alpha \in p\} \in \mathbb{N}.\] If however \(J\) is infinite, the dimension of \(Z\) (defined in the “obvious” way) may be infinite (e.g. \(\dim(K^{J})\) is infinite). We need to define when \(Z'\) is a large subset of \(Z\) even when \(\dim(Z)\) is infinite.
Definition 16. Let \(Y \subseteq Z\) be subsets of \(K^{J}\) which are type-definable (over some small \(A \subset K\)). We say that \(Y\) is large in \(Z\) if, for every \(J' \subseteq J\) with \(J'\) finite, \(\dim(\Pi_{J'}(Y)) = \dim(\Pi_{J'}(Z))\).
Given a partial type \(p(x_J)\), we denote by \(p(\mathbb{K})\) the set of realizations of \(p\) in \(\mathbb{K}^{J}\), and \(\dim(p(x_J)) \mathrel{\vcenter{:}}= \dim(p(\mathbb{K})\)). Given \(J' \subseteq J\), we denote by \(p(J')\) the partial type corresponding to the set \(\Pi_{J'}(p(\mathbb{K}))\).
Let \(q \in S^{J}(A)\) be a complete type, of dimension \(d\).
Lemma 49. Assume that \(J\) is finite.
Then, there is a family \(\{X_{i}: i \in I\}\) of \(L(A)\)-definable sets such that:
\(q(\mathbb{K}) = \bigcap_{i \in I} X_{i}\);
each \(X_{i}\) has constant local dimension \(d\).
Proof. Let \(\{Y_{i}: i \in I\}\) be a family of \(L(A)\)-definable sets such that: \(q(\mathbb{K}) = \cap \{Y_{i}: i \in I\}\). Let \(Z\) be some \(L(A)\)-definable set of dimension \(d\) containing \(q(\mathbb{K})\): by replacing \(Y_{i}\) with \(Y_{i} \cap Z\), w.l.o.g. we may assume that all \(Y_{i}\) have dimension \(d\). Given an \(L(A)\)-definable set \(Y \subseteq K^{J}\) of dimension \(d\), let \[\mathop{\mathrm{Sing}}(Y) \mathrel{\vcenter{:}}= \{\bar a\in Y: \mathop{\mathrm{loc-dim}}_{\bar a}(Y) < d\}\\and\mathop{\mathrm{Reg}}(Y) \mathrel{\vcenter{:}}= Y \setminus \mathop{\mathrm{Sing}}(Y).\] Since \(\dim\) is definable, \(\mathop{\mathrm{Sing}}(Y)\) and \(\mathop{\mathrm{Reg}}(Y)\) are \(L(A)\)-definable. Since \(\dim\) is local, \(\dim(\mathop{\mathrm{Sing}}(Y)) < d\), and therefore \(\dim(\mathop{\mathrm{Reg}}(Y)) = d\) and \(\mathop{\mathrm{Reg}}(Y)\) has constant local dimension \(d\).
Claim 6. For every \(i \in I\), \(q(\mathbb{K}) \subseteq \mathop{\mathrm{Reg}}(Y_{i})\).
Assume not: then, since \(q\) is a complete type over \(A\), \(q(\mathbb{K}) \subseteq \mathop{\mathrm{Sing}}(Y_{i})\), but then \(\dim(q) \leq \dim(\mathop{\mathrm{Sing}}(Y_{i})) < d\), absurd.
Thus, we can define \(X_{i} \mathrel{\vcenter{:}}= \mathop{\mathrm{Reg}}(Y_{i})\). ◻
Notice that in the above lemma we used that \(\dim\) is definable.
Lemma 50. Let \(U \subseteq K^{J}\) be open and definable (with parameters in some small set \(B\) with \(A \subseteq B \subset K\)). If \(U \cap q(\mathbb{K}^{J})\) is nonempty, then it is large inside \(q(\mathbb{K}^{J})\).
Proof. Since \(U\) is definable, there is some \(J_{0} \subseteq J\) finite and \(V \subseteq K^{J_{0}}\) \(L(A)\)-definable and open, such that \(U = \pi_{J_{0}}^{-1}(V)\). Let \(J_{0} \subseteq J' \subseteq J\), with \(J'\) finite. We need to prove that \(\Pi_{J'}(U \cap q(K^{J'}))\) is large inside \(\Pi_{J'}(q(\mathbb{K}))\).
The former is equal to \(U' \cap q(\mathbb{K}(J'))\), where \(U' \mathrel{\vcenter{:}}= \Pi_{J'}(U')\): thus, by replacing \(U\) with \(U'\), \(J\) with \(J'\), and \(q(x_{J})\) with \(q(x_{J'})\), we may assume that \(J'\) is finite. Thus, by 49, we may assume that \(q(\mathbb{K}) = \bigcap_{i}X_{i}\), where each \(X_{i}\) is an \(L(A)\)-definable set of constant local dimension \(d\). Thus, since \(X_{i} \cap U\) is nonempty, \(X_{i} \cap U\) has (local) dimension \(d\), and therefore \(\dim(\bigcap _{i} X_{i} \cap U) = d\). ◻
Proof of Theorem 47. W.l.o.g., we may assume that \(\langle \mathbb{K}, \bar \delta \rangle\) is a monster model and \(A\) is “small”. By Beth definability, it suffices to show that \(X\) is \(L(A)\)-invariant (i.e., that \(X\) is set-wise invariant under automorphisms of \(\mathbb{K}\) as \(L\)-structure fixing \(A\) point-wise).
Let \(Z \mathrel{\vcenter{:}}= \mathop{\mathrm{Jet}}(X) \subseteq K^{\Gamma_{n}}\) and \(Y\) be the closure of \(Z\) (according to the topology \(\Pi_{\mathbb{K}, \Gamma_{n}}\)). Since \(X\) is closed, we have that \(X = \Pi_{n}(Y)\) (where \(\Pi_{n}\) is the projection onto the first \(n\) coordinates). Thus, it suffices to show that \(Y\) is \(L(A)\)-invariant.
Let \(\bar a, \bar a' \in K^{\Gamma_{n}}\), with the same \(L(A)\)-type, and such that \(\bar a\in Y\). We need to show that \(\bar a' \in Y\). Since \(\tau_{\mathbb{K}, n}\) is definable and \(Y\) is closed, it suffices to prove the following:
Claim 7. Let \(U' \subseteq K^{\Gamma_{n}}\) be an \(L(\mathbb{K})\)-definable open set containing \(\bar a'\). Then, \(U'\) intersects \(Y\).
Let \(\phi\) be an \(L(A)\)-automorphism of \(\mathbb{K}\) such that \(\phi(\bar a') = \bar a\). Let \(U \mathrel{\vcenter{:}}= \phi(U')\): notice that \(U\) is an open \(L(A)\)-definable set containing \(\bar a\); therefore, there exists \(\bar b\in X\) such that \(\mathop{\mathrm{Jet}}(\bar b) \in U\). Let \(\bar c'\mathrel{\vcenter{:}}= \phi^{-1}(\mathop{\mathrm{Jet}}(\bar b))\).
Let \(q(\bar x_{\Gamma})\) be the \(L(A)\)-type of \(\mathop{\mathrm{Jet}}(\bar b)\): by definition, it is also the \(L(A)\)-type of \(\bar c'\). By 50, \(q(\mathbb{K}) \cap U'\) is large inside \(q(\mathbb{K})\). Thus, by the Extension Theorem, there exists \(\bar b' \in K^{n}\) such that \[\mathop{\mathrm{Jet}}(\bar b') \in q(\mathbb{K}) \cap U'.\] Therefore, \(\bar b' \in X\) and therefore \(\mathop{\mathrm{Jet}}(\bar b') \in Z \cap U' \subseteq Y \cap U'\). ◻
Let \(\langle A, \bar \delta \rangle\) be a field (of characteristic \(0\)) with \(k\) commuting derivations. The derivations induce a matroid on \(A\). Given \(a \in A\), \(Y \subseteq A\) and \(X\subseteq A\), we define \(a \in \mathop{\mathrm{\bar \delta-acl}}_{Y}(X)\) if \(\mathop{\mathrm{Jet}}(a)\) is not algebraically independent over \(\mathop{\mathrm{Jet}}(X \cup Y)\).
As shown in [@FK], \(\mathop{\mathrm{\bar \delta-acl}}_{Y}\) is a matroid on \(A\).6
Fix \(\langle \mathfrak C, \bar \delta \rangle\) monster model of \(T^{\bar \delta}_g\). We have the corresponding matroid \(\mathop{\mathrm{\bar \delta-acl}}(X) \mathrel{\vcenter{:}}= \mathop{\mathrm{\bar \delta-acl}}_{F}(X)\).
Theorem 51. \(\mathop{\mathrm{\bar \delta-acl}}\) is an existential matroid (in the sense of [@Fornasiero:matroid]).
Proof. We have to prove that \(\mathop{\mathrm{\bar \delta-acl}}\) is definable and it satisfies existence (see [@Fornasiero:matroid]).
The fact that \(\mathop{\mathrm{\bar \delta-acl}}\) is definable means that, for every \(A \subseteq \mathfrak C\) and \(b \in \mathop{\mathrm{\bar \delta-acl}}(A)\) there exists an \(L^{\bar \delta}\)-formula \(\phi(\bar x, y)\) and \(\bar a\in A^{n}\) such that \(\langle \mathfrak C, \bar \delta \rangle \models \phi(\bar a,b)\) and, for every \(\bar a',b'\) in \(\mathfrak C\), if \(\langle \mathfrak C, \bar \delta \rangle \models \phi(\bar a', b')\), then \(b' \in \mathop{\mathrm{\bar \delta-acl}}(\bar a')\). We can take as \(\phi\) any formula witnessing that \(b^{\mathop{\mathrm{Jet}}}\) is not algebraically independent over \(A\).
For existence, let \(A \subseteq B \subset \mathfrak C\) be subsets of small cardinality. Let \(c \in \mathfrak C\) such that \(c \notin \mathop{\mathrm{\bar \delta-acl}}(A)\). We have to show that there exists \(d \in \mathfrak C\) such that \(c\) and \(d\) have the same \(L^\delta\)-type over \(A\) and \(d \notin \mathop{\mathrm{\bar \delta-acl}}(B)\).
Since \(\makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}\) satisfies existence, there exists \(d \in \mathfrak C\) such that \(c\) and \(d\) have the same \(L^\delta\)-type over \(A\) and \(d \makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}_{A} B\). Then, \(\mathop{\mathrm{Jet}}(d)\) is algebraically independent over \(\mathop{\mathrm{Jet}}(B)\): therefore, \(d \notin \mathop{\mathrm{\bar \delta-acl}}(B)\), proving that \(\mathop{\mathrm{\bar \delta-acl}}\) is an existential matroid. ◻
Thus, \(\mathop{\mathrm{\bar \delta-acl}}\) induces a dimension function \(\mathop{\mathrm{\bar \delta-\!\dim}}\) on models of \(T^{\bar \delta}_g\) (see [@Fornasiero:matroid]; see also [@GP:12]).
Remark 52.
\(\mathop{\mathrm{\bar \delta-acl}}\) is not the \(T^{\bar \delta}_g\)-algebraic closure: the former only contains the latter. For instance, the whole field of constants \(\mathcal{C}_{\bar \delta}\) is in \(\mathop{\mathrm{\bar \delta-acl}}(\emptyset)\).
\(\makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}\) is not the independence relation induced by \(\mathop{\mathrm{\bar \delta-acl}}\), because \(\makebox[1.7\width][l]{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M, \bar \delta}}\,\,\,\,}}}\) is strict. For instance, if \(a \in \mathcal{C}_{\bar \delta}\setminus \mathop{\mathrm{acl}}(F)\), then \(a \mathrel{\not\mkern-7mu{\mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textrm{M,\bar\delta}}\,\,\,\,}}}}_{\emptyset} a\).
Lemma 53 (See [@ELR]). Let \(\langle \mathbb{K}, \bar \delta \rangle \models T^{\bar \delta}_g\). Let \(Y \subseteq K^{n}\) be \(L\)-definable (with parameters). Then, \(\dim(Y) = \mathop{\mathrm{\bar \delta-\!\dim}}(Y)\).
Proof. By the properties of dimension functions (see [@Dries:89]) it suffices to treat the case when \(n = 1\) (the general case follows by induction on \(n\)). If \(\dim(X) = 0\), then \(X\) is finite, and therefore \(\mathop{\mathrm{\bar \delta-\!\dim}}(X) = 0\). If \(\dim(X) = 1\), then \((X - X)/(X - X) = \mathbb{K}\), and therefore \(\mathop{\mathrm{\bar \delta-\!\dim}}(X) = 1\). ◻
The same proof gives a more general result.
Proposition 54 (Invariance of dimension for fields). Let \(L\) be a language expanding the language of rings, and \(L^{*}\) be an expansion of \(L\). Let \(A^{*}\) be an \(L^{*}\)-structure expanding a field, and \(A\) be its restriction to the language \(L\). Assume that \(\dim^{*}\) and \(\dim\) be dimension functions on \(A^{*}\) and \(A\), respectively. Then, for every \(X \subseteq A^{n}\) which is \(L\)-definable (with parameters), \(\dim^{*}(X) = \dim(X)\).
Unlike in the case of lovely pairs, we cannot approximate \(L^\delta\)-definable sets with \(L\)-definable sets.
Remark 55. Let \(\langle \mathbb{K}, \delta \rangle \models T^{\delta}_g\). Let \(X \subseteq K\) be \(L^\delta\)-definable (with parameters). If \(X\) is definable in the lovely pair \(\langle \mathbb{K}, \mathcal{C}_{\bar \delta} \rangle\) (see 10), then there exists \(Y \subseteq K\) which is \(L\)-definable and such that \(\mathop{\mathrm{\bar \delta-\!\dim}}(X \Delta Y) < 1\) ([@Fornasiero:matroid]). If not, such \(Y\) might not exist: for instance, let \(\mathbb{K}\) be a real closed field, and \(X \mathrel{\vcenter{:}}= \{x \in K: \delta x > 0\}\).
The assumption that the derivations commute cannot be dropped.
Lemma 56. If the derivations do not commute, then \(\mathop{\mathrm{\bar \delta-acl}}_{Y}\) is not a matroid, because it is not transitive.7 In fact, let \(k = 2\) and \(\langle \mathbb{K}, \bar \delta \rangle \models T^{\bar \delta,nc}_g\). Then, there exist \(a, b, c \in K\) such that:
\(a^{\Gamma}\) is algebraically independent over \(F\);
\(\delta_{2} b = 0\) and \(\delta_{1} b = \delta_{1} a\);
\(c = a - b\).
Notice that \(\delta_{1}c = 0\). Then, \(a \notin \mathop{\mathrm{\bar \delta-acl}}(F)\), \(b,c \in \mathop{\mathrm{\bar \delta-acl}}(F)\), but \(a \in \mathop{\mathrm{\bar \delta-acl}}_{F}(b,c)\): thus, transitivity fails.
Lemma 57. For \(k\geq 2\), models of \(T^{\bar \delta,nc}_g\) do not have a dimension function.
Proof. For simplicity, we do the case when \(k = 2\). Let \(\langle \mathbb{K}, \bar \delta \rangle \models T^{\bar \delta,nc}_g\). Assume, by contradiction, that \(\dim'\) is a dimension function on \(\langle \mathbb{K}, \bar \delta \rangle\). Let \(X \mathrel{\vcenter{:}}= \{b \in K: \delta_{1} b = 0\}\). Let \(Y \mathrel{\vcenter{:}}= \{c \in K: \delta_{2} c = 0\}\). Notice that \(X\) and \(Y\) are \(L^\delta\)-definable subfields of \(K\) of infinite index inside \(\mathbb{K}\): thus, \(\dim'(X) = \dim'(Y) = 0\).
Claim 8. \(X + Y = K\).
Let \(a \in K\). Let \(b \in K\) such that \(\delta_{1} b = 0\) and \(\delta_{2} b = \delta_{2} a\), and let \(c = a - b\). Notice that \(b \in X\) and \(c \in Y\). Thus \(X + Y = K\).
But \(\dim'(X) = \dim'(Y) = 0\), and therefore \(\dim'(K) = 0\), while the axioms of dimension require that \(\dim'(K) = 1\). ◻
We denote by \(K^{K}\) the set of all functions from \(K\) to \(K\), and by \(\mathop{\mathrm{Der_{K}}}\subset K^{K}\) the set of derivations on \(K\) extending \(\eta\) (remember that \(\eta\) is a fixed derivation on the field \(F = dcl(\emptyset)\): see §2.1). The main references for this section is [@Hjorth], from which our presentation is heavily inspired; for the background notions of descriptive set theory see [@Kechris].
For every \(\bar a, \bar b\in K^{n}\), we define \[B_{\bar a, \bar b} \mathrel{\vcenter{:}}= \{\delta \in K^{K}: \delta(\bar a) = \bar b\}.\] For every \(L^\delta\)-sentence \(\phi\) with parameters in \(\mathbb{K}\), we define \[U_{\phi} \mathrel{\vcenter{:}}= \{\delta \in K^{K}: \langle \mathbb{K}, \delta \rangle \models \phi\}.\] The set \(K^{K}\) has two “canonical” topologies:
The pro-discrete topology, whose basis of open sets is given by \[\{B_{\bar a,\bar b}: \bar a, \bar b\in K^{n}, n \in \mathbb{N}\},\] and which we denote by \(\tau_d\), which is the topology induced by the product topology on \(K^{K}.\)
The “first-order” topology, whose basis of open sets is given by \[\{U_{\phi}: \phi \text{ L^\delta-sentence with parameters in \mathbb{K}}\},\] and which we denote by \(\tau_{FO}\).
Remark 58. Another basis for \(\tau_d\) is \[\{U_{\phi}: \phi \text{ quantifier-free L^\delta-sentence with parameters in \mathbb{K}}\}\]
In fact, \[B_{\bar a,\bar b} = U_{(\delta a_{1} = b_{1} \wedge \dotsb \wedge \delta a_{n} = b_{n})}.\]
For the remainder of this section, when we don’t specify the topology, we mean \(\tau_d\). The following notions concerning specific formulas are useful for obtaining information about sets with respect to the topology.
Definition 17. We say that an \(L^\delta\)-sentence \(\phi\) with parmeters \(\bar a\) is “relatively quantifier free” if \(\phi = \alpha(\mathop{\mathrm{Jet}}_{\delta}^{}(\bar a))\) for some \(L\)-formula without parameters \(\alpha\).
We say \(\phi\) is “relatively existential” if \[\phi = \exists \bar x\;\alpha(\mathop{\mathrm{Jet}}(\bar a), \mathop{\mathrm{Jet}}(\bar x)),\] for some \(L\)-formula without parameters \(\alpha\).
We can define “relatively universal” and “relatively \(\forall\exists\)” \(L^\delta\)-sentences with parametrs.
Lemma 59. Let \(\phi\) be an \(L^\delta\) sentence with parameters.
If \(\phi\) is relatively quantifier free, then \(U_{\phi}\) is clopen.
If \(\phi\) is relatively existential, then \(U_{\phi}\) is open.
If \(\phi\) is relatively universal, then \(U_{\phi}\) is closed.
If \(\phi\) is relatively \(\forall\exists\), then \(U_{\phi}\) is \(\mathcal{G}_{\delta}\).
Proof. We do only the case when \(\phi\) is relatively existential: the others are similar. Write \(\phi = \exists \bar y\;\alpha(\bar a, \delta \bar a, \dotsc, \delta^{n}\bar a, \bar b, \delta \bar b, \dotsc, \delta^{m} \bar b)\), for some \(L\)-formula \(\phi\). Then, \[\begin{gather} U_{\phi} = \bigcup \bigl( B_{\bar a,\bar a_{1}} \cap B_{\bar a_{1},\bar a_{2}} \cap \dotsb \cap B_{\bar a_{n-1},\bar a_{n}} \cap B_{\bar b,\bar b_{1}} \cap B_{\bar b_{1},\bar b_{2}} \cap \dotsb \cap B_{\bar b_{m-1},\bar b_{m}}:\\ \bar a_{1}, \dotsc, \bar a_{n}, \bar b_{1}, \dotsc, \bar b_{m} \in K^{< \omega} \wedge \langle \mathbb{K}, \delta \rangle \models \alpha(\bar a, \bar a_{1}, \dotsc, \bar a_{n}, \bar b, \bar b_{1}, \dotsc, \bar b_{m}) \bigr). \end{gather}\] ◻
For the remainder of this section, we assume that \(\mathbb{K}\) and \(L\) are countable.
Thus, \(\mathop{\mathrm{Der_{K}}}\) is \(\tau_{FO}\)-closed and it is a \(\tau_d\)-\(\mathcal{G}_{\delta}\) inside \(K^{K}\); we use the same names for the induced topologies on \(\mathop{\mathrm{Der_{K}}}\). Notice that \(K^{K}\) is a Polish space: therefore, \(\mathop{\mathrm{Der_{K}}}\) is also a Polish space (see [@Kechris]). Thus, any two dense \(\mathcal{G}_{\delta}\) subsets of \(\mathop{\mathrm{Der_{K}}}\) always intersect.
Given \(Z \subseteq K^{n} \times K^{n}\), we define \[I_{Z} \mathrel{\vcenter{:}}= \{ \delta \in \mathop{\mathrm{Der_{K}}}: \exists \bar b\in K^{n}: \langle \bar b,\delta \bar b \rangle \in Z \}\]
Lemma 60. For every \(Z \subseteq K^{n} \times K^{n}\), \(I_{Z}\) is an open subset of \(\mathop{\mathrm{Der_{K}}}\).
Proof. \[I_{Z} = \bigcup\bigl( B_{\bar a,\bar b} : \bar a\in K^{n}, \bar b\in K^{n}, \langle \bar a, \bar b \rangle \in Z \bigr).\] ◻
Let \(\mathbb{G}\) be the family of derivations \(\delta \in \mathop{\mathrm{Der_{K}}}\) such that \(\langle \mathbb{K}, \delta \rangle \models T^{\delta}_g\). Let \(\mathcal{L}\) be the family of the sets \(Z \subseteq K^{n + n}\) definable with parameters, such that \(\Pi_{n}(Z)\) is large (for some \(n \in \mathbb{N}\)).
Lemma 61. \(\mathbb{G}(M) = \bigcap_{Z \in \mathcal{L}} I_{Z}\). Moreover, \(\mathbb{G}\) is a \(\mathcal{G}_{\delta}\)-subset of \(\mathop{\mathrm{Der_{K}}}\).
Proof. By the axiomatization \({T^{\delta}_{\mathrm{wide}}}\) (where we introduced in [@FT:24] ), \(\mathbb{G}(M) = \bigcap_{Z \in \mathcal{L}} I_{Z}\). Each \(I_{Z}\) is open. By our assumptions, \(\mathcal{L}\) is countable. ◻
Lemma 62. On \(\mathbb{G}\), \(\tau_{FO}\) and \(\tau_d\) coincide.
Proof. By elimination of quantifiers, every \(L^\delta\)-sentence is equivalent, modulo \(T^{\delta}_g\), to a relatively quantifier-free sentence. The conclusion follows from 59. ◻
Lemma 63. Assume that \(\mathop{\mathrm{rk}}(K / F)\) is infinite. Then, for every \(\bar a\) finite tuple in \(K\) and every \(W\) large subset of \(K^{n}\) which is \(L\)-definable with parameters, there exists \(\bar b\in W\) which is algebraically independent over \(F\bar a\).
Proof. By induction on \(n\), it suffices to treat the case when \(n = 1\). Let \(b \in K \setminus \mathop{\mathrm{acl}}(F\bar a)\). Since \(W \subseteq K\) is large, then there exists \(b_1, b_2, b_3, b_4 \in W\) such that \((b_{1} - b_{2})/(b_{3} - b_{4}) = b\) and \(b_3 \not = b_4\). Therefore, at least one of the \(b_{i}\) is not in \(\mathop{\mathrm{acl}}(F\bar a)\). ◻
Theorem 64. There exists \(\mathbb{K}\models T\) which is countable and of infinite rank over \(F\). For any such \(\mathbb{K}\), the set \(\mathbb{G}\) is a dense subset of \(\mathop{\mathrm{Der_{K}}}\).
Thus, in a precise topological sense, \(\mathbb{G}\) is a generic set (notice that \(\mathbb{G}\) is \(\tau_{FO}\)-closed in \(\mathop{\mathrm{Der_{K}}}\)).
Proof. We have seen that each \(I_{Z}\) is open. It suffices to prove the following claim.
Claim 9. For every \(Z \in \mathcal{L}\), \(I_{Z}\) is dense.
Let \(Z \subseteq K^{n} \times K^{n}\). Let \(B_{\bar a, \bar b}\) be a nonempty basic open set. We have to verify that \(I_{Z} \cap B_{\bar a, \bar b}\) is nonempty. Let \(\varepsilon\in B_{\bar a, \bar b}\): that is, \(\varepsilon\in \mathop{\mathrm{Der_{K}}}\) and \(\varepsilon\bar a= \bar b\). Let \(\varepsilon_{0}\) be the restriction of \(\varepsilon\) to \(\mathop{\mathrm{acl}}(F\bar a)\). Let \(\bar c\in \Pi_{n}(Z)\) be algebraically independent over \(F\bar a\). We can extend \(\varepsilon_{0}\) arbitrarily to \(\bar c\); in particular, there exists \(\delta \in \mathop{\mathrm{Der_{K}}}\) such that \(\delta\) extends \(\varepsilon_{0}\) and \(\delta \bar c\in Z\). Thus, \(\delta \in I_{Z} \cap B_{\bar a, \bar b}\). ◻
The following theorem gives a “topological” criterion for when a differential system has a solution in models of \(T^{\delta}_g\).
Theorem 65. Let \(\langle \mathbb{K}, \varepsilon \rangle \models T^{\delta}\). Assume that \(\mathbb{K}\) countable and of infinite rank over \(F\). Let \(\bar a\in K^{\ell}\). Let \(\mathop{\mathrm{Der_{K}}}(\bar a, \varepsilon)\) be the set of derivations \(\delta\) on \(K\) extending \(\eta\) and such that \(\varepsilon\) and \(\delta\) coincide on \(\mathop{\mathrm{Jet}}_{\epsilon}(\bar a)\) where it is the Jet related to the derivation \(\epsilon.\)
Let \(Z \subseteq K^{n} \times K^{n}\) be \(L\)-definable with parameters \(\bar a\). Let \(\delta\) be a derivation in \(K\) such that \(\langle \mathbb{K}, \delta \rangle \models T^{\delta}_g\). T.f.a.e.:
\(I_{Z}\) is dense in \(\mathop{\mathrm{Der_{K}}}(\bar a, \varepsilon)\);
\(I_Z\) is nonempty;
\(I_{Z} \cap \mathbb{G}\) is nonempty;
\(\delta \in I_{Z}\).
Proof. First of all it is easy to see that \((1) \implies (2),\) and \((4) \implies (3)\) are obvious. Moreover (2) is equivalent to (3), since \(I_{Z}\) by Lemma 60 is open and \(\mathbb{G}\) by Theorem 64 is dense in \(\mathop{\mathrm{Der_{K}}}\).
We prove first the case when \(\bar a\) is empty (that is, \(Z\) is \(L\)-definable without parameters and \(\mathop{\mathrm{Der_{K}}}(\bar a, \varepsilon) = \mathop{\mathrm{Der_{K}}}\)).
In this case, we can add another equivalent formulation to [en:generic-4]:
Since \(T^{\delta}_g\) is complete, and “\(\delta \in I_{Z}\)” can be expressed as a first-order sentence (without parameters), we have that \((4) \implies (5),\) the converse is trivial so (4), (5) are equivalent. Therefore, (1) is equivalent to (2).
Let us consider now the case when \(\bar a\) is non-empty. Let \(F' \mathrel{\vcenter{:}}= F[\mathop{\mathrm{Jet}}_{\epsilon}(\bar a)]\) and let \(\eta'\) be the restriction of \(\varepsilon\) to \(F'\). We denote by \(L' \mathrel{\vcenter{:}}= L(F)\), and \(T' \mathrel{\vcenter{:}}= T \cup \mathop{\mathrm{Diag}}(F')\).
We can consider the theory \({T'}^{\delta}_{g}\) of generic derivations on \(K\) extending \(\eta'\): notice that \(\langle \mathbb{K},\delta \rangle \models {T'}^{\delta}_{g}\). We can apply the previous proof to \(T'\), since \(Z\) is now \(L'\)-definable in \(K\) without parameters (notice that we need to modify the definition of \(\mathbb{G}\), since we are restricting the space of derivations to those extending \(\eta'\): however, we already proved the equivalence between (2) and (3)). We conclude in this way the proof. ◻
Barbina and Zambella [@BZ] deal with a similar situation: however, we cannot use their results, since to apply them to our setting we would need that \(\mathbb{K}\) is countable and saturated. Maybe there could be a common refinement if one could weaken their assumption to \(\mathbb{K}\) resplendent (since every countable consistent theory has a countable resplendent model: see [@Hodges]).
In the article [@FT:dexp], we posed several conjectures. We now conclude the paper with a list of additional open problems, remarks, and some further ideas.
Let \(\langle \mathbb{K}, \bar \delta \rangle\models T^{\bar \delta,?}_g\). Given a type \(p \in S_{L^{\bar \delta}}^{n}(\mathbb{K})\), let \(\bar a\) be a realization of \(p\); we define \(\tilde{p} \in S_{L}^{\Gamma_n}(\mathbb{K})\) as the \(L\)-type of \(\mathop{\mathrm{Jet}}(\bar a)\) over \(\mathbb{K}\).
::: open problem Open problem 66. Is it true that \(p\) is definable iff \(\tilde{p}\) is definable? We conjecture that it is true when \(T^{\bar \delta,?}_g= T^{\bar \delta}_g\) and \(T\) has NIP. :::
Given \(X \subseteq K^{n}\), denote by \(X^{Zar}\) be the Zariski closure of \(X\).
Questions 67 (See [@FLL]). 1) Let \(\bigl( X_{i}: i \in I \bigr)\) be an \(L\)-definable family of subsets of \(K^{n}\). Is \(\bigl( X_{i}^{Zar}: i \in I \bigr)\) also \(L\)-definable?
2) Assume that 1) holds for \(\mathbb{K}\). Let \(\langle \mathbb{K}, \bar \delta \rangle \models T^{\bar \delta,?}_g\). Let \(\bigl( X_{i}: i \in I \bigr)\) be an \(L^\delta\)-definable family of subsets of \(K^{n}\). Is \(\bigl( X_{i}^{Zar}: i \in I \bigr)\) also \(L^\delta\)-definable?
We have affirmative answers to both questions, which will be presented in a paper in preparation.
Let \(\langle \mathfrak C, \bar \delta \rangle\) be a monster model of \(T^{\bar \delta}_g\). Let \(\bar a\in \mathfrak C^{n}\), \(B \subseteq \mathfrak C\) such that \(\bar \delta B \subseteq B\). There exists a polynomial \(\omega_{\bar a\mid B}(t)\) such that, for \(n\) large enough, \(\mathop{\mathrm{rk}}(\mathop{\mathrm{Jet}}_n(\bar a) \mid B) = \omega_{\bar a\mid B}(n)\), where \(\mathop{\mathrm{Jet}}_n(\bar a) = \{\mu \bar a: \mu \in \Gamma and \lvert\mu \rvert\leq n\}\) (see [@Kolchin]). The degree of the polynomial is at most \(k\); denote by \(\mu(\bar a)\) the leading monomial of \(\omega_{\bar a\mid K}\) (including its coefficient). Let \(X \subseteq K^{n}\) be \(L^{\bar \delta}\)-definable with parameters \(\bar b\): define \[\begin{align} \mu(X) &\mathrel{\vcenter{:}}= \sup \bigl( \mu(\bar a\mid {\mathop{\mathrm{Jet}}}(\bar b)): \bar a\in X \bigr)\\ \omega_{X} &\mathrel{\vcenter{:}}= \sup \bigl( \omega_{\bar a\mid {\mathop{\mathrm{Jet}}(\bar b)}}: \bar a\in X \bigr). \end{align}\] where the supremum for \(\omega\) is taken inside \(\mathbb{R}[t]\) w.r.t. the order \(p>q\) if \(\lim_{t \to + \infty} p(t) > q\) (cf.[@FLL]), while the supremum of \(\mu\) is taken w.r.t. the order \(r t^d \leq r' t^{d'}\) if \(d < d'\) or \(d = d'\) and \(r \leq r'\) (cf.[@Fornasiero:Hilbert-length]). Notice that, by 6, \(\mu(X)\) and \(\omega_{X}\) are well-defined (that is, they not depend on the choice of the parameters \(\bar b\)).
Conjecture 68 (See [@FLL; @Riviere:09]). The suprema in the definitions of \(\omega_X\) and \(\mu(X)\) are maxima. Moreover, \(\omega\) and \(\mu\) are definable in families: that is, for every \(L^{\bar \delta}\)-definable family \(\bigl( X_{i}: i \in I \bigr)\) there exists a partition of \(I\) into finitely many definable set \(I = I_{1} \sqcup \dotsb \sqcup I_{m}\) such that \(\mu(X_{i})\) and \(\omega_{X_{i}}\) are constant on each \(I_{j}\).
::: open problem Open problem 69. What is the “geometric” meaning of \(\mu(X)\)? Notice that, up to a multiplicative constant, the \(k^{th}\) coefficent of \(\omega_{X}\) is equal to \(\mathop{\mathrm{\bar \delta-\!\dim}}(X)\). :::
If 68 is true, then the function \(X \mapsto \mu(X)\) behaves like a dimension on \(L^\delta\)-definable sets (with the difference that the values of \(\mu\) are not natural numbers, but monomials: cf.[@Dries:89]).
Conjecture 70. Assume that \(\mathfrak C\) is endowed with a topology \(\tau\) satisfying some suitable conditions. Let \(\tau_{\bar \delta}\) be the topology on \(\mathfrak C\) induced by the embedding \(\mathfrak C\to \mathfrak C^{\Gamma}\), \(x \mapsto \mathop{\mathrm{Jet}}(x)\) (where \(\mathfrak C^{\Gamma}\) is endowed with the product topology induced by \(\tau\)). Denote by \(\overline{X}^{\tau_{\bar \delta}}\) the \(\tau_{\bar \delta}\)-closure of \(X\). Then, for every \(X \subseteq \mathfrak C^{n}\) which is \(L^{\bar \delta}\)-definable and nonempty, \(\overline{X}^{\tau_{\bar \delta}}\) is also \(L^{\bar \delta}\)-definable, and \(\mu(\overline{X}^{\tau_{\bar \delta}}\setminus X) < \mu(X)\).
Acknowledgements The authors thank Elliot Kaplan, Itay Kaplan, Noa Lavi, Silvain Rideau-Kikuchi, and Marcus Tressl. We thank the anonymous referee for the careful reading of the manuscript and for the constructive suggestions.
Funding Both authors are members of the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA-INDAM). The authors acknowledge financial support from INdAM (Istituto Nazionale di Alta Matematica), particularly through the Intensive Period in Model Theory, “Model Theory and tame expansions of topological Fields”, Naples, 19 May–18 July 2025.
Later, we will prove a kind of converse to this fact.↩︎
When \(\mathbb{K}\) is algebraically bounded, the above definition coincides with the one in 4.↩︎
\(T^{\bar \delta,?}_g\) is differentially algebraically bounded: see [@Wang:25]; cf.[@Dries:89].↩︎
This step was not clear to us: thanks to Itay Kaplan for explaining it.↩︎
Notice that in their proof [@MS] do not mention that \(\mathfrak C\) should have IACS: but this holds in their setting. [@dElbee:23a] gives a similar result that might be easier to prove in this context.↩︎
A more general result is true. Let \(\mathbb{A} := \langle A, \mathop{\mathrm{cl}} \rangle\) be a finitary matroid. Let \(\bar \delta\) be a tuple of commuting quasi-endomorphisms of \(\mathbb{A}\), in the sense of [@FK]. Given \(X, Y \subseteq A\), define \(\bar \delta\)-\(cl_{Y}(X)\) as the set of \(a \in A\) such that \(a^{\mathop{\mathrm{Jet}}}\) is not \(\mathop{\mathrm{cl}}\)-independent over \(\mathop{\mathrm{Jet}}(X)\mathop{\mathrm{Jet}}(Y)\). Then, \(\bar \delta\)-\(\mathop{\mathrm{cl}}\) is a finitary matroid on \(A\).↩︎
Naturally, we use the free monoid \(\Gamma\) instead of the free commutative monoid \(\mathop{\mathrm{Jet}}\) to define \(\mathop{\mathrm{\bar \delta-acl}}\) in this situation.↩︎