Must every nonalgebraic type in a countable, strictly stable, NDOP, shallow theory be nonorthogonal to a regular type?

Laskowski and Shelah's trichotomy theorem for countable, strictly stable (stable but not superstable) theories states that every such theory satisfies at least one of the following: it has the dimensional order property (DOP); it has NDOP but is deep; or it admits an abelian group witness to unsuperstability, a descending sequence of definable abelian groups with infinite successive indices whose connected intersection has regular generic type. In the NDOP-and-shallow case, the structural corner of strict stability relevant to their attempted main gap for $\aleph_1$ -saturated models, the paper poses its main open question: must every nonalgebraic complete type of such a theory be nonorthogonal to some regular type? In superstable theories this 'enough regular types' property is classical and underlies the regular-type decomposition machinery of Shelah's classification program; whether it survives strict stability is unknown. The authors prove that a counterexample, if one exists, must contain an abelian group witness whose intersection has generic type both $(\mathcal{ID}_r)^\perp$ -minimal and foreign to the regular ideal $\mathcal{ID}_r=\{\varphi : r\perp\varphi\}$ attached to the offending type $r$ . The question remains open: the announced companion paper on the Karp complexity of stable theories never appeared, and the closest later work (Palacín-Shelah 2018 on flat theories) proves only partial nonorthogonality results and explicitly leaves the general question unresolved.

Background

A complete first-order theory $T$ is stable if no formula linearly orders an infinite sequence of tuples; equivalently, for every set $A$ of parameters the set $S(A)$ of complete types over $A$ has size at most $|A|^{|T|}$ . In a stable theory there is a canonical notion of independence called forking: a type forks over a set $A$ if it implies a finite disjunction of formulas each of which divides over $A$ , i.e., such that no single tuple satisfies $k$ of its instances along a long $A$ -indiscernible sequence of parameter tuples, for some finite $k$ . Two stationary types $p$ and $q$ are orthogonal, written $p\perp q$ , when any realization of $p$ and any realization of $q$ are forking-independent over every common base over which both types are based; nonorthogonality is the fundamental relation along which stability theory organizes types. A type is nonalgebraic if it has infinitely many realizations, i.e., is not isolated by a formula $x=a$ .

A stationary type is regular if it is orthogonal to all of its forking extensions. Regular types carry a pregeometry, and hence a dimension, on their sets of realizations, and they are the building blocks of Shelah's structure theory for superstable theories. A stable theory is superstable if every type does not fork over some finite subset of its domain, and strictly stable (stable but unsuperstable) otherwise. A classical theorem says that a superstable theory has enough regular types: every nonalgebraic type is nonorthogonal to a regular type. This fact underlies the regular decomposition of types and, combined with the structural hypotheses below, the main gap theorems of classification theory; the existence of regular types in superstable theories is analyzed in Shelah-Buechler (1989). Whether 'enough regular types' persists in any class of strictly stable theories is a basic structural question and is the subject of this problem.

For strictly stable theories much less structure is known. The dimensional order property (DOP) is a nonstructure property of stable theories; its negation, NDOP, is the hypothesis that permits every sufficiently saturated model to be decomposed as an a-prime model over an independent tree of smaller saturated models (an 'a-saturated' model is one realizing every complete type over each of its parameter subsets of size below $\kappa_r(T)$ , the least cardinal not occurring as the length of a forking chain in $T$ — for countable strictly stable $T$ this bound is $\aleph_1$ — and an 'a-prime' model over a set is a prime model within that saturation class). A stable theory is deep if there is a sequence of a-saturated models $\langle M_n : n<\omega\rangle$ with $\operatorname{tp}(M_{n+1}/M_n)\perp M_{n-1}$ for all $n\ge 1$ , so that decompositions have unbounded depth, and shallow otherwise. For superstable theories, NDOP together with shallowness marks the classifiable regime where the structure theorems hold.

Laskowski and Shelah (arXiv:0711.3043, published as Trans. Amer. Math. Soc. 363 (2011) 1619-1629) proved the analogous trichotomy for the strictly stable world: every complete theory in a countable language that is stable and unsuperstable satisfies at least one of the following: it has DOP; it has NDOP but is deep; or it admits an abelian group witness to unsuperstability, meaning a descending sequence $\langle A_n : n<\omega\rangle$ of definable abelian groups with each index $[A_n:A_{n+1}]$ infinite, whose intersection $A=\bigcap_n A_n$ is connected (has no definable subgroup of finite index) and whose generic type is regular — indeed, the trichotomy supplies such a witness whose intersection has generic type both $\mathbf{R}^\infty$ -minimal and foreign to $\mathbf{R}^\infty$ . Here the generic type of a connected definable group in a stable theory is the distinguished nonforking 'random element' type of the group. Their proof develops the notion of a regular ideal of formulas: a class $\mathcal{ID}$ of formulas with parameters, containing all algebraic formulas and closed under automorphisms, finite disjunction, and downward entailment, satisfying a regularity clause (if $\psi\in\mathcal{ID}$ and $\theta(x,b)\in\mathcal{ID}$ for every $b$ satisfying $\psi$ , then $\exists y(\psi(y)\wedge\theta(x,y))\in\mathcal{ID}$ ). The central example is $\mathbf{R}^\infty$ , the ideal of superstable formulas. A type is $\mathcal{ID}$ -small if it entails some formula of $\mathcal{ID}$ , and foreign to $\mathcal{ID}$ if it is orthogonal to every $\mathcal{ID}$ -small type; a type is $(\mathcal{ID})^\perp$ -minimal if it is not $\mathcal{ID}$ -small yet every forking extension obtained by a formula whose parameter type is foreign to $\mathcal{ID}$ is $\mathcal{ID}$ -small. A key lemma of the paper is that a type both $(\mathcal{ID})^\perp$ -minimal and foreign to $\mathcal{ID}$ is regular.

In the NDOP-and-shallow case of the trichotomy, which is the setting of the authors' attempted 'main gap for $\aleph_1$ -saturated models,' the abelian group witness is automatically present, and the paper poses its main open question: is every nonalgebraic type of such a theory nonorthogonal to a regular type? Their Theorem 4.4 recasts the question: if a nonalgebraic stationary type $r$ orthogonal to every regular type exists in such a theory, then the theory contains an abelian group witness whose intersection has generic type both $(\mathcal{ID}_r)^\perp$ -minimal and foreign to $\mathcal{ID}_r$ , where $\mathcal{ID}_r=\{\varphi : r\perp\varphi\}$ is the regular ideal attached to $r$ . Since that generic type is itself regular (by the lemma above) and is orthogonal to $r$ , a counterexample must realize a delicate configuration in which the offending type coexists with a specific regular-type witness of unsuperstability.

The question also gates a descriptive-complexity program. In earlier work the same authors measured the 'Karp complexity' of a class $\mathcal{K}$ of structures by controlledness: for every cardinal $\lambda$ , the relation of $L_{\infty,\lambda}$ -equivalence, equivalence in the infinitary logic permitting conjunctions over arbitrary sets and quantification over $\lambda$ -tuples (by Karp's theorem the same as back-and-forth equivalence), partitions $\mathcal{K}$ into a set rather than a proper class of equivalence classes; they proved that pseudo-elementary classes (classes of reducts, to a smaller language, of models of a first-order theory in an expanded language) containing a formula with the independence property (a formula that, for a suitable infinite set of tuples, traces out every subset of that set by a choice of parameters) are never controlled. The companion paper 'The Karp complexity of stable theories,' cited in the trichotomy paper as in preparation, was to apply the trichotomy to classes of uncountable models of stable theories; it has never appeared, so the nonorthogonality question stands as the unresolved structural input for both applications.

The closest later progress on 'enough regular types' is Palacín-Shelah (2018), who introduce $\omega$ -forking (a strengthening of forking) and the flat theories, the stable theories in which every type fails to $\omega$ -fork over some finite set, a class containing the superstable theories. They prove that every type has a non- $\omega$ -forking extension that is nonorthogonal to a regular type, and that in flat theories every type is nonorthogonal to a type of weight one (a type that cannot be split into two independent forking pieces), while stating explicitly that they cannot ensure in general that every type is nonorthogonal to a regular one.

Problem Statement

Let $T$ be a complete theory in a countable language that is stable but not superstable, has NDOP (the negation of the dimensional order property), and is shallow (not deep). Must every nonalgebraic complete type of $T$ be nonorthogonal to some regular type of $T$ ?

Equivalently: does there exist such a theory $T$ together with a nonalgebraic stationary type $r$ (over some set of parameters) that is orthogonal to every regular type of $T$ ?

Laskowski and Shelah proved (Theorem 4.4 of the preprint arXiv:0711.3043, in the section presenting the paper's second application; theorem numbering follows the preprint, whose maintained arXiv source postdates the 2011 journal version) that if such a type $r$ exists, then $T$ contains an abelian group witness to unsuperstability, a descending sequence $\langle A_n : n<\omega\rangle$ of definable abelian groups with $[A_n:A_{n+1}]$ infinite, whose intersection $A=\bigcap_n A_n$ has generic type that is both $(\mathcal{ID}_r)^\perp$ -minimal and foreign to $\mathcal{ID}_r$ , where $\mathcal{ID}_r=\{\varphi\in L(\mathfrak{C}) : r\perp\varphi\}$ . Thus a negative answer must realize this specific configuration, while a positive answer must hold for every theory in the stated class, with no restriction on the language beyond countability, on models or cardinalities, or on the form of the answer.

References

Michael C. Laskowski, Saharon Shelah, 'Karp height of models of stable theories' (preprint of the paper published as 'A trichotomy of countable, stable, unsuperstable theories'; the PDF's internal title is 'A nonstructure theorem for countable, stable, unsuperstable theories'), arXiv:0711.3043, 2007, https://arxiv.org/abs/0711.3043 (LKM source paper)
Michael C. Laskowski, Saharon Shelah, 'A trichotomy of countable, stable, unsuperstable theories', Transactions of the American Mathematical Society 363(3), 2011, pp. 1619-1629, DOI 10.1090/S0002-9947-2010-05196-7, https://doi.org/10.1090/S0002-9947-2010-05196-7
Daniel Palacín, Saharon Shelah, 'On the class of flat stable theories', Annals of Pure and Applied Logic 169(8), 2018, pp. 835-849, DOI 10.1016/j.apal.2018.04.005, arXiv:1801.01438, https://arxiv.org/abs/1801.01438
Saharon Shelah, Steven Buechler, 'On the existence of regular types', Annals of Pure and Applied Logic 45(3), 1989, pp. 277-308, DOI 10.1016/0168-0072(89)90039-0, https://doi.org/10.1016/0168-0072(89)90039-0
Michael C. Laskowski, Saharon Shelah, 'The Karp complexity of unstable classes', Archive for Mathematical Logic 40(2), 2001, pp. 69-88, DOI 10.1007/s001530000047, arXiv:math/0011167, https://arxiv.org/abs/math/0011167
Michael C. Laskowski, Saharon Shelah, 'Karp complexity and classes with the independence property', Annals of Pure and Applied Logic 120(1-3), 2003, pp. 263-283, DOI 10.1016/S0168-0072(02)00080-5, arXiv:math/0303345, https://arxiv.org/abs/math/0303345
Michael C. Laskowski, Saharon Shelah, 'Decompositions of saturated models of stable theories', Fundamenta Mathematicae 191(2), 2006, pp. 95-124, DOI 10.4064/fm191-2-1, https://doi.org/10.4064/fm191-2-1
Elisabeth Bouscaren, Bradd Hart, Ehud Hrushovski, Michael C. Laskowski, 'Non-locally modular regular types in classifiable theories', arXiv:1910.11404, 2019 (v3 2024, unpublished as of 2026-08-30), https://arxiv.org/abs/1910.11404