Does uniqueness of limit models imply mu-symmetry or mu^+-symmetry in superstable abstract elementary classes?

An abstract elementary class (AEC) is a semantic generalization of the class of models of a complete first-order theory, and a central problem of its classification theory is to identify the right notion of superstability. VanDieren (2016) proved that in a mu-superstable AEC with amalgamation and joint embedding, mu-symmetry - a symmetry property of Shelah's non-mu-splitting dependence relation on Galois types - implies that limit models of cardinality mu are unique (and saturated), that mu^+-symmetry transfers down to mu-symmetry, and that mu^+-symmetry makes unions of increasing chains of mu^+-saturated models mu^+-saturated. The same paper asks the converse: does the purely structural statement that limit models of cardinality mu are unique already force mu^+-symmetry, or even mu-symmetry, in mu-superstable classes? Under tameness the structural and independence-theoretic sides are known to be equivalent (Grossberg-Vasey), and all positive uniqueness results for limit models through 2025 still assume a symmetry-type hypothesis; neither a ZFC proof nor a counterexample is known without locality assumptions on Galois types. The problem is posed as a prove-or-refute question about the exact strength of uniqueness of limit models relative to symmetry of non-splitting.

Background

An abstract elementary class (AEC) is a pair $\mathcal{K}=(K,\prec_{\mathcal{K}})$ consisting of a class of structures closed under isomorphism and a strong substructure relation $\prec_{\mathcal{K}}$, satisfying coherence ($M\prec_{\mathcal{K}}N$ implies $M\subseteq N$, and $M_0\subseteq M_1\prec_{\mathcal{K}}N$ with $M_0\prec_{\mathcal{K}}N$ implies $M_0\prec_{\mathcal{K}}M_1$), closure under $\prec_{\mathcal{K}}$-increasing chains and their intersections, and a Lowenheim-Skolem number $\mathrm{LS}(\mathcal{K})$: every member has a $\prec_{\mathcal{K}}$-substructure of size at most $\mathrm{LS}(\mathcal{K})$ containing any prescribed small subset. The class of models of a complete first-order theory with elementary substructure is the basic example, but AECs need not be axiomatizable, so classification-theoretic arguments must replace syntax with semantic notions.

In place of syntactic types one uses Galois (orbital) types: $\mathrm{ga\text{-}tp}(a/M,N)$ is the orbit of the tuple $a$ under automorphisms of a sufficiently large ambient domain fixing the model $M$ pointwise; $\mathrm{ga\text{-}S}(M)$ is the set of Galois types over $M$. $\mathcal{K}$ is Galois-stable in $\mu$ if $|\mathrm{ga\text{-}S}(M)|\leq\mu$ for every $M\in\mathcal{K}_\mu$, where $\mathcal{K}_\mu$ denotes the members of $\mathcal{K}$ of cardinality $\mu$.

The basic forking-like dependence relation available in general AECs is Shelah's splitting: a type $p\in\mathrm{ga\text{-}S}(N)$ does not $\mu$-split over $M\prec_{\mathcal{K}}N$ (with $M\in\mathcal{K}_\mu$) if for all $N_1,N_2\in\mathcal{K}_\mu$ with $M\prec_{\mathcal{K}}N_\ell\prec_{\mathcal{K}}N$ and every isomorphism $f:N_1\to N_2$ fixing $M$ pointwise, one has $f(p\upharpoonright N_1)=p\upharpoonright N_2$.

A model $N$ is universal over $M$ (of the same cardinality $\mu$) if every $N'\in\mathcal{K}$ with $M\prec_{\mathcal{K}}N'$ and $\|N'\|\leq\mu$ embeds into $N$ over $M$. For a limit ordinal $\theta<\mu^+$, $N$ is a $(\mu,\theta)$-limit model over $M_0$ if $N=\bigcup_{i<\theta}M_i$ for an increasing continuous chain $\langle M_i\mid i\leq\theta\rangle$ in $\mathcal{K}_\mu$ with $M_0$ the base and $M_{i+1}$ universal over $M_i$. Limit models exist under Galois-stability in $\mu$ together with no maximal models (a maximal model is a member of $\mathcal{K}$ with no proper $\prec_{\mathcal{K}}$-extension in $\mathcal{K}$), and they play the role that saturated models - models realizing every type over every parameter set of smaller cardinality - play in first-order theory. Uniqueness of limit models of cardinality $\mu$ means: for all limit ordinals $\theta_1,\theta_2<\mu^+$ and any two $(\mu,\theta_1)$- and $(\mu,\theta_2)$-limit models $M_1,M_2$ over the same base $N\in\mathcal{K}_\mu$, $M_1$ and $M_2$ are isomorphic over $N$.

Following the source paper, $\mathcal{K}$ is $\mu$-superstable if it is Galois-stable in $\mu$ and non-$\mu$-splitting has local character along chains of universal extensions of limit models: for every infinite ordinal $\alpha$, every increasing sequence $\langle M_i\mid i<\alpha\rangle$ of limit models of cardinality $\mu$ with $M_{i+1}$ universal over $M_i$, and every $p\in\mathrm{ga\text{-}S}(\bigcup_{i<\alpha}M_i)$, there is $i<\alpha$ such that $p$ does not $\mu$-split over $M_i$. This is the local analogue of the first-order local character of forking.

$\mu$-symmetry (symmetry for non-$\mu$-splitting), introduced by VanDieren, is the property: whenever $M,M_0,N\in\mathcal{K}_\mu$ and elements $a,b$ satisfy (1) $M$ is universal over $M_0$ and $M_0$ is a limit model over $N$; (2) $a\in M\setminus M_0$; (3) $\mathrm{ga\text{-}tp}(a/M_0)$ is non-algebraic (not realized by any element of the model over which the type is taken) and does not $\mu$-split over $N$; and (4) $\mathrm{ga\text{-}tp}(b/M)$ is non-algebraic and does not $\mu$-split over $M_0$; then there exists a limit model $M^b$ over $M_0$ containing $b$ such that $\mathrm{ga\text{-}tp}(a/M^b)$ does not $\mu$-split over $N$. It is the AEC analogue of symmetry of first-order forking and is equivalent (in $\mu$-superstable classes) to the statement that reduced towers - certain increasing sequences of models with designated elements and submodels coding non-splitting - are continuous at limit stages.

The classical first-order fact motivating the program is that superstability is equivalent to the assertion that the union of any increasing chain of saturated models is saturated. For AECs there are several non-equivalent candidates for superstability (Shelah's superlimit models - models isomorphic to every $\prec_{\mathcal{K}}$-extension of the same cardinality - good frames - Shelah's axiomatization of superstable independence in a local frame - no long splitting chains - a bound on the lengths of increasing chains of models each splitting over its predecessor - $\mu$-superstability, and uniqueness of limit models), and how they relate is a driving question of AEC classification theory. The source paper proves, for AECs satisfying the amalgamation property (any two members with a common $\prec_{\mathcal{K}}$-substructure embed over it into a common member) and the joint embedding property (any two members embed into a common member), with no maximal models of cardinality $\mu^+$ and $\mu\geq\mathrm{LS}(\mathcal{K})$: Theorem 2, $\mu^+$-symmetry implies $\mu$-symmetry when $\mathcal{K}$ is $\mu$- and $\mu^+$-superstable; Theorem 15, $\mu$-superstability plus $\mu$-symmetry implies uniqueness of limit models of cardinality $\mu$, and these limit models are saturated; Theorem 1, $\mu^+$-symmetry plus $\mu$- and $\mu^+$-superstability implies that unions of increasing chains of $\mu^+$-saturated models are $\mu^+$-saturated (here saturated means Galois-saturated: the model realizes every Galois type over each of its $\prec_{\mathcal{K}}$-submodels of cardinality at most $\mu$, respectively $\mu^+$); and Corollary 18, in a class with arbitrarily large models, categoricity in $\mu^{+n}$ for some finite $n\geq 1$ implies $\mu$-symmetry. The converse arrows are missing, and the paper asks whether uniqueness of limit models alone suffices.

What is known about the converse: tameness - a locality property asserting that Galois types agreeing on all restrictions to $\kappa$-sized parameter sets are equal - closes the gap. Grossberg and Vasey proved that in tame AECs with arbitrarily large models, uniqueness of limit models is asymptotically equivalent to no long splitting chains, existence of superlimit models, solvability (Shelah's structural stand-in for stability, expressed by the class having a skeleton closed under certain amalgamation and uniqueness), and the union-of-saturated-models property; VanDieren and Vasey proved that superstability plus tameness implies symmetry, and separately transferred symmetry downward across a whole interval of cardinals $[\mu,\lambda]$ without tameness or set-theoretic assumptions. But without any locality of Galois types no route from uniqueness back to symmetry is known, and no counterexample is known either: every class in which uniqueness of limit models is known to hold (first-order superstable classes, tame classes, categorical classes) also has symmetry.

Problem Statement

Let $\mathcal{K}$ be an abstract elementary class satisfying the amalgamation property and the joint embedding property, with no maximal models of cardinality $\mu^+$ (that is, every member of $\mathcal{K}$ of cardinality $\mu^+$ has a proper $\prec_{\mathcal{K}}$-extension in $\mathcal{K}$), and let $\mu\geq\mathrm{LS}(\mathcal{K})$. Suppose $\mathcal{K}$ is $\mu$-superstable and $\mu^+$-superstable, and that $\mathcal{K}$ has uniqueness of limit models of cardinality $\mu$: for all limit ordinals $\theta_1,\theta_2<\mu^+$ and all $M_1,M_2,N\in\mathcal{K}_\mu$ such that $M_\ell$ is a $(\mu,\theta_\ell)$-limit model over $N$, $M_1$ and $M_2$ are isomorphic over $N$. These standing hypotheses are exactly the context of the implication diagram (Figure 1) in the source paper under which the question below is posed; the assumption of $\mu^+$-superstability is included because part (a) concerns $\mu^+$-symmetry and needs structure at $\mu^+$ to be non-vacuous.

(a) Must $\mathcal{K}$ satisfy $\mu^+$-symmetry, that is, symmetry for non-$\mu^+$-splitting (the property stated in the background with $\mu$ replaced by $\mu^+$)?

(b) Must $\mathcal{K}$ satisfy $\mu$-symmetry, that is, symmetry for non-$\mu$-splitting?

Both questions are to be settled in ZFC, without tameness or any other locality property of Galois types, and without large-cardinal or cardinal-arithmetic assumptions. Since $\mu^+$-symmetry implies $\mu$-symmetry under the stated superstability hypotheses (Theorem 2 of the source paper), a positive answer to (a) implies a positive answer to (b), so (b) is the weaker target; a negative answer to (b) refutes both.

References

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