Countable models of weakly quasi-o-minimal theories II
We confirm Martin's conjecture for a broad subclass of weakly quasi-o-minimal theories.
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arXiv subjects
Slavko Moconja
Publications and source records attributed to Slavko Moconja.
Countable models of weakly quasi-o-minimal theories I
We introduce the notions of triviality and order-triviality for global invariant types in an arbitrary first-order theory and show that they are well behaved in the NIP context. We show that these two notions agree for invariant global extensions of a weakly o-minimal type, in which case we say that the type is trivial. In the o-minimal case, we prove that every definable complete 1-type over a model is trivial. We prove that the triviality has several favorable properties; in particular, it is preserved in nonforking extensions of a weakly o-minimal type and under weak nonorthogonality of weakly o-minimal types. We introduce the notion of a shift in a linearly ordered structure that generalizes the successor function. Then we apply the techniques developed to prove that every weakly quasi-o-minimal theory that admits a definable shift has $2^{\aleph_0}$ countable models.
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Weakly o-minimal types
We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type $p\in S(A)$ is weakly o-minimal if for some relatively $A$-definable linear order, $<$, on $p(\mathfrak{C})$ every relatively $L_{\mathfrak{C}}$-definable subset of $p(\mathfrak{C})$ has finitely many convex components in $(p(\mathfrak{C}),<)$. We establish many nice properties of weakly o-minimal types. For example, we prove that weakly o-minimal types are dp-minimal and share several properties of weight-one types in stable theories, and that a version of monotonicity theorem holds for relatively definable functions on the locus of a weakly o-minimal type.
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Invariant measures in simple and in small theories
We give examples of (i) a simple theory with a formula (with parameters) which does not fork over the empty set but has mu measure 0 for every automorphism invariant Keisler measure mu, and (ii) a definable group G in a simple theory such that G is not definably amenable, i.e. there is no translation invariant Keisler measure on G We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups, and nontriviality of the graded Grothendieck ring.
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Ramsey theory and topological dynamics for first order theories
We investigate interactions between Ramsey theory, topological dynamics, and model theory. We introduce various Ramsey-like properties for first order theories and characterize them in terms of the appropriate dynamical properties of the theories in question (such as [extreme] amenability of a theory or some properties of the associated Ellis semigroups). Then we relate them to profiniteness and triviality of the Ellis groups of first order theories. In particular, we find various criteria for [pro]finiteness and for triviality of the Ellis group of a given theory from which we obtain wide classes of examples of theories with [pro]finite or trivial Ellis groups. As an initial motivation, we note that profiniteness of the Ellis group of a theory implies that the Kim-Pillay Galois group of this theory is also profinite, which in turn is equivalent to the equality of the Shelah and Kim-Pillay strong types. We also find several concrete examples illustrating the lack of implications between some fundamental properties. In the appendix, we give a full computation of the Ellis group of the theory of the random hypergraph with one binary and one 4-ary relation. This example shows that the assumption of NIP in the version of Newelski's conjecture for amenable theories (proved in [15]) cannot be dropped.
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Does weak quasi-o-minimality behave better than weak o-minimality?
We present a relatively simple description of binary, definable subsets of models of weakly quasi-o-minimal theories. In particular, we closely describe definable linear orders and prove a weak version of the monotonicity theorem. We also prove that weak quasi-o-minimality of a theory with respect to one definable linear order implies weak quasi-o-minimality with respect to any other such order.
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Around Rubin's "Theories of linear order"
Let $\mathcal M=(M,<,...)$ be a linearly ordered first-order structure and $T$ its complete theory. We investigate conditions for $T$ that could guarantee that $\mathcal M$ is not much more complex than some colored orders (linear orders with added unary predicates). Motivated by Rubin's work, we label three conditions expressing properties of types of $T$ and/or automorphisms of models of $T$. We prove several results which indicate the "geometric" simplicity of definable sets in models of theories satisfying these conditions. For example, we prove that the strongest condition characterizes, up to definitional equivalence (inter-definability), theories of colored orders expanded by equivalence relations with convex classes.
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Stationarily ordered types and the number of countable models
We introduce notions of stationarily ordered types and theories; the latter generalizes weak o-minimality and the first is a relaxed version of weak o-minimality localized at the locus of a single type. We show that forking, as a binary relation on elements realizing stationarily ordered types, is an equivalence relation and that each stationarily ordered type in a model determines some order-type as an invariant of the model. We study weak and forking non-orthogonality of stationarily ordered types, show that they are equivalence relations and prove that invariants of non-orthogonal types are closely related. The developed techniques are applied to prove that in the case of a binary, stationarily ordered theory with fewer than $2^{\aleph_0}$ countable models, the isomorphism type of a countable model is determined by a certain sequence of invariants of the model. In particular, we confirm Vaught's conjecture for binary, stationarily ordered theories.
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Asymmetric regular types
We study asymmetric regular types. If $\frak p$ is regular and $A$-asymmetric then there exists a strict order such that Morley sequences in $\frak p$ over $A$ are strictly increasing (we allow Morley sequences to be indexed by elements of a linear order). We prove that for all $M\supseteq A$ maximal Morley sequences in $\frak p$ over $A$ consisting of elements of $M$ have the same (linear) order type, denoted by $\Inv_{\frak p,A}(M)$, which does not depend on the particular choice of the order witnessing the asymmetric regularity. In the countable case we determine all possibilities for $\Inv_{\frak p,A}(M)$: either it can be any countable linear order, or in any $M\supseteq A$ it is a dense linear order (provided that it has at least two elements). Then we study relationship between $\Inv_{\frak p,A}(M)$ and $\Inv_{\frak q,A}(M)$ when $\frak p$ and $\frak q$ are strongly regular, $A$-asymmetric, and such that $\frak p_{\strok A}$ and $\frak q_{\strok A}$ are not weakly orthogonal. We distinguish two kinds on non-orthogonality: bounded and unbounded. In the bounded case we prove that $\Inv_{\frak p,A}(M)$ and $\Inv_{\frak q,A}(M)$ are either isomorphic or anti-isomorphic. In the unbounded case, $\Inv_{\frak p,A}(M)$ and $\Inv_{\frak q,A}(M)$ may have distinct cardinalities but we prove that their Dedekind completions are either isomorphic or anti-isomorphic. We provide examples of all four situations.
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