Model Companion of an Endomorphism

• Model companion of an endomorphism is a canonical structure ensuring existential closedness in algebraic systems with a designated endomorphism.
• It generalizes classical frameworks by using kernel configurations and geometric axioms to control definability and generic behavior.
• Key properties include uniqueness, quantifier elimination, and compatibility with o-minimal open core, benefiting analysis in fields and vector spaces.

A model companion of an endomorphism is a fundamental concept in model theory, describing the canonical existentially closed structure within classes of algebraic systems expanded by a distinguished endomorphism. This paradigm provides a robust framework to analyze generic behaviors of endomorphisms in rich algebraic contexts, such as fields or vector spaces, through first-order methods. Model companions articulate axioms that, informally, ensure the lack of "unexpected" algebraic or definable constraints—generalizing the notion of a generic automorphism or endomorphism from classical settings into a unified algebraic landscape.

1. Foundational Frameworks and Core Languages

The construction of a model companion of an endomorphism begins by extending a base theory—typically an equational theory of a field or vector space—with a new unary function symbol $\theta$ intended as a (multiplicative or linear) endomorphism. For example, in the field case, the language is

$L = \{ +, \cdot, -1, 0, 1; \theta \}$

and the base theory $T_0$ asserts that $\theta$ is a multiplicative map: $\theta(xy) = \theta(x) \theta(y)$, $\theta(1) = 1$, and $\theta(x) = 0 \Leftrightarrow x = 0$ (d'Elbée, 2022). For vector spaces, $T_\theta$ ensures, in every model $M$ of a complete, model-complete theory $T$, that a definable infinite $K$-vector space $V(M)$ carries $\theta$ as a $K$-linear endomorphism; formally, $$T_\theta := T \cup \{ \theta|_V \ \text{is} \ K\text{-linear}, \forall x \notin V: \theta(x) = 0 \}$$ (Chini, 19 Feb 2025, Chini, 20 Dec 2025).

To capture various algebraic behaviors of $\theta$, the approach generalizes to selecting "kernel configurations" $C$ that encode constraints on polynomial expressions in $\theta$ via identities of the form: $$\sum_k \bigcap_l \ker \bigl( \rho_{j,k,l}[\theta] \bigr) = \sum_k \bigcap_l \ker \bigl( \eta_{j,k,l}[\theta] \bigr)$$ over systems of polynomials $\rho, \eta \in K[X]$ (Chini, 19 Feb 2025).

2. Model Companions: Existence, Uniqueness, and Axiomatics

In such extended languages, the model companion—if it exists—provides a model-complete theory described by axioms ensuring "generic" behavior of the endomorphism. In the context of fields (the "ACFH" theory), the model companion of algebraically closed fields with a multiplicative endomorphism is uniquely characterized by a geometric axiom scheme. This scheme requires that for every tuple of minimal multiplicative equations and for any constructible, multiplicatively free set, suitable lifting properties between tuples and their $\theta$-images are realized: $$\forall \varphi, \tau, d, c \quad \left( \text{multiplicative freeness} \ \Rightarrow \ \exists u,v \; \varphi(u, v, \theta(u), \theta(v), d) \right)$$ ensuring existential closedness among all extensions where the structure of $(K, \theta)$ is preserved [(d'Elbée, 2022), Theorem 2.7].

For vector spaces, the model companion for each kernel configuration $C$ (denoted $T\theta^C$) is determined by

• $C$-image-completeness: For every $f$ with $C(f) < \infty$, the images of $f^{C(f)}[\theta]$ and $f^{C(f)+1}[\theta]$ coincide.
• Power Nullstellensatz: For any bounded quantifier-free formula and certain system of linear equations subject to $C$, existential liftings of solutions are always found internally (Chini, 19 Feb 2025, Chini, 20 Dec 2025).

These axioms are expressed in first-order form precisely when the base theory $T$ satisfies a uniform definability of independent extensions (often labeled “H$_4$” following d'Elbée), ensuring the existence and axiomatizability of the model companions (Chini, 19 Feb 2025).

3. Classification: Structures, Completions, and Quantifier Elimination

The resulting model companions admit precise structural descriptions:

• In ACFH, every completion is unique and isomorphic over the underlying algebraically closed field data, reflecting uniqueness as a model-completion [(d'Elbée, 2022), Theorem 2.12].
• For vector spaces, completions are classified up to the isomorphism type of the $\theta$-closure of the empty set (i.e., the smallest substructure closed under $\theta$ and algebraic operations) [(Chini, 20 Dec 2025), Corollary 5.2].
• Quantifier elimination can be achieved in an expanded language that includes all definable endomorphisms $R_C$ determined by the kernel configuration, yielding a uniform quantifier elimination analogous to that in module theory or difference fields, depending on $R_C$ [(Chini, 20 Dec 2025), Theorem 5.5].

In the pure vector space case with $T$ the theory of $K$-vector spaces, $T_{K,\theta}^C$ is simply the theory of $R_C$-modules.

All definable sets in $T\theta^C$ are described, up to logical equivalence, by disjunctions of existential quantifications over “algebraic patterns”—ladders of equations built from the definable endomorphism ring and ordinary $L$-formulas algebraic in the variables [(Chini, 20 Dec 2025), Theorem 4.1].

4. Model-Theoretic Properties: Independence, Simplicity, and Imaginaries

Model companions of endomorphism theories display distinguishing model-theoretic features:

• NSOP$_1$ and non-simplicity: In ACFH, every completion is NSOP$_1$ but not simple. Kim-independence is characterized via a ternary relation involving the $\theta$-closure of parameter sets; failure of base monotonicity excludes simplicity [(d'Elbée, 2022), Theorem 3.12].
• Algebraic closure: In $T\theta^C$, the $L_\theta$-algebraic closure $\operatorname{acl}_{L_\theta}(A)$ is generated by iterating algebraic closure in $L$ and all $R_C$-endomorphisms—a linear closure if $R_C$ is a field, yielding strong minimality with exchange [(Chini, 20 Dec 2025), Theorem 5.7].
• Elimination of imaginaries: In ACFH, if forking satisfies the existence axiom, elimination of imaginaries follows, as per Conant--Kruckman criteria adapted to the NSOP$_1$ setting [(d'Elbée, 2022), Theorem 3.17].

5. Structure and Genericity of Kernels

A salient feature of model companions for endomorphism theories is the structure of the kernels of definable endomorphisms:

• Generic pseudo-finite abelian groups: In ACFH, for any nonzero $P \in \mathbb{Z}[X]$, the kernel of $P(\theta)$, $\ker P(\theta)$, is a generic multiplicative subgroup of $K$ and is pseudo-finite-cyclic as a pure abelian group. Specifically, this means it is elementarily equivalent to an ultraproduct of finite cyclic groups, characterized by the condition $|G[p]| = |G/pG| \leq p$ for every prime $p$ [(d'Elbée, 2022), Theorem 4.8].
• Infinite kernels and ultraproduct obstruction: No non-principal ultraproduct of natural finite-kerneled maps yields a model of ACFH. In all ACFH models, the kernel is necessarily infinite, precluding realization as a limit of maps like $x \mapsto x^n$ in finite fields [(d'Elbée, 2022), Lemma 4.13].

For vector spaces, existentially closed models corresponding to a configuration $C$ enforce completeness conditions on the images of iterates $\rho[\theta]$ that are canonical in module theory and directly generalize the field case (Chini, 19 Feb 2025).

6. Variants, Examples, and Extensions

The formalism of kernel configurations $C$ allows a taxonomy of model companions for endomorphism theories, subsuming diverse behaviors:

Kernel configuration $C$	Defining Constraint	Nature of the Companion Theory
$C_0$: all zeros	All nonzero $\rho[\theta]$ injective	Theory of generic injective endomorphisms
$C_\infty$: all infinities	No extra kernel constraint	Theory with generic unconstrained endomorphism
$C_{f, r}$: $c(f) = r$	$f^r[\theta]$ full image; $f^{r+1}[\theta]$ not	Companion imposing given kernel multiplicity
$C$ algebraic, MiPo$(C) = X^2-2$	$\theta^2 = 2 \operatorname{Id}$	Satisfies “power Nullstellensatz”-style axiom

(Chini, 19 Feb 2025, Chini, 20 Dec 2025).

In o-minimal settings, if $T$ is o-minimal and the vector space structure is continuous, the expansion $T\theta^C$ does not preserve o-minimality globally, but possesses o-minimal open core: every definable open set is already definable in the reduct to $T$ (with parameters in the $\theta$-closure) [(Chini, 20 Dec 2025), Theorem 6.2]. This ensures that the added complexity from $\theta$ does not introduce pathological topological phenomena in tame geometry.

7. Connections and Implications

The study of model companions of endomorphisms generalizes the model theory of difference fields (as in the ACFA context, Chatzidakis–Hrushovski 1999) while introducing perspective on how general unary operators interact with ambient algebraic or geometric structures. The kernel-based “configuration” parameterization unifies disparate cases—nilpotent, injective, or arbitrary polynomially constrained endomorphisms—under a single model-theoretic paradigm.

A plausible implication is that any future axiomatization or classification involving operators in an algebraic context can likely be situated within the kernel-configuration and existential-completeness frameworks constructed in these works. The open core phenomenon further suggests compatibility between deep notions of definability in topological/geometric model theory and the combinatorial constraints imposed by generic endomorphism actions.

Key references for foundational methods and analogies include (d'Elbée, 2022, Chini, 19 Feb 2025, Chini, 20 Dec 2025).