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Predrag Tanović

Publications and source records attributed to Predrag Tanović.

13 recordsLinked to original sources
arXiv2026

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.

math.LO↗
arXiv2026

The number of countable models of first-order theories

Throughout, $T$ denotes a complete first-order theory in a countable language $L$ that has infinite models and $I(\aleph_0,T)$ denotes the number of countable models of $T$, up to an isomorphism. To determine $I(\aleph_0,T)$, it suffices to consider only countable models of $T$ with domain $ω$; since there are at most continuum many $L$-structures with domain $ω$, $I(\aleph_0,T)\leqslant 2^{\aleph_0}$ holds. Theories with $I(\aleph_0,T)=1$ are the $\aleph_0$-categorical theories. These include the theory of an infinite set, theories of infinite-dimensional vector spaces over a finite field, and the theory of dense linear orders. Theories with $I(\aleph_0,T)<2^{\aleph_0}$ are said to have few countable models. In this paper we discuss and survey work done on Vaught's conjecture, Martin's conjecture, and Ehhrenfeuch theories (theories with more than one but only finitely many, countable models).

math.LO↗
arXiv2026

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.

math.LO↗
arXiv2021

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.

math.LO↗
arXiv2021

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.

math.LO↗
arXiv2019

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.

math.LO↗
arXiv2013

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.

math.LO↗
arXiv2013

Generically stable regular types

We study non-orthogonality of symmetric, regular types and show that it preserves generic stability and is an equivalence relation on the set of all generically stable, regular types. We will also prove that some of the nice properties from the stable context hold in general. In the case of strongly regular types we will relate non-orthogonality to the global Rudin-Keisler order.

math.LO↗
arXiv2013

Simple groups and the number of countable models

Let $T$ be a complete, superstable theory with fewer than $2^{\aleph_{0}}$ countable models. Assuming that generic types of infinite, simple groups definable in $T^{eq}$ are sufficiently non-isolated we prove that $ω^ω$ is the strict upper bound for the Lascar rank of $T$.

math.LO↗
arXiv2012

Around Podewski's conjecture

A long-standing conjecture of Podewski states that every minimal field is algebraically closed. It was proved by Wagner for fields of positive characteristic, but it remains wide open in the zero-characteristic case. We reduce Podewski's conjecture to the case of fields having a definable (in the pure field structure), well partial order with an infinite chain, and we conjecture that such fields do not exist. Then we support this conjecture by showing that there is no minimal field interpreting a linear order in a specific way; in our terminology, there is no almost linear, minimal field. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is elementary abelian of prime exponent. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is torsion.

math.LO↗